Chapt1 Infinitude of Twin Primes proof by correcting Regular Primes #1 Correcting Math 3rd ed.
Archimedes Plutonium
developed into this
3rd edition.
Index:
Chapters
(1) Proof of Infinitude of Twin Primes by correcting Regular Primes
proof
(2) other proofs of primes from correction of Indirect Method on
Regular Primes
(3) proof of no odd perfect number and the infinitude of perfect
numbers
(4) proof of the Goldbach Conjecture
(5) Correction of 4 Mapping with its convoluted borders counted or not
counted
(6) a new system of infinite-integers, called AP-adics to escape base
dependency of
Hensel p-adics
(7) Precision Definition of Finite versus Infinite
(8) why the Poincare Conjecture is always false
(9) proof of Kepler Packing Problem where the most dense packing is a
modification
of hexagonal closed pack
(10) why the Riemann Hypothesis is not a mathematics conjecture
(11) precision definitions of Sequence, Series, Function, Continuity,
Algebra
(12) Old-Math from Pythagoras to 2010 enters into the New-Math era
The key chapter of this book is chapter 7 of the precision definition
of Finite versus
Infinite as given by a borderline between finite and infinity at
10^603.
This first chapter is mostly about the correction of a logic error
that most
mathematicians used when doing the Indirect method proof of Euclid
Infinitude of Primes.
This is important mainly because there is only one proof of the
Infinitude of Twin Primes
that is possible in mathematics. Mathematics has a lot of proofs that
have only one
proof method and an indirect method for that proof. The proof of
Infinitude of Regular Primes (IP) has very many methods and has both
the direct and indirect forms of proving IP. But some proofs in
mathematics have only one proof method, and perhaps one can say that a
majority of proofs of mathematics are amenable to just one singular
method of proving. If
we look to geometry given the axioms, we usually find there is only
one method to build a
proof for that statement.
As it so happens, for Twin Primes, mathematics has only one method of
proving it. And that
method happens to be the Indirect or Reductio Ad Absurdum method. Some
call it the method of contradiction. So the importance of having a
valid proof by Indirect for Euclid's Infinitude
of Regular Primes can be appreciated, because if that proof is not
valid, then it is the source of
inability to ever give a proof of Twin Primes.
The Twin Primes conjecture was stated in the time of Euclid in Ancient
Greek times.
And no-one has ever come close to proving the infinitude of twin
primes, until 1991.
By making a observation of the Indirect method proof of Euclid's
Infinitude of Regular Primes
it is noticed that the Euclid Number must be necessarily a new prime.
That is the key to
generating a proof of the Infinitude of Twin Primes. And the reason no
mathematician saw the
flaw of the indirect proofs of Euclid IP, is because of a lack of
logic concerning a irrelevant
detail of 1+2x3x5x7x11x13 = 59x509. To a person, not well inclined to
logical necessity, they
can easily stumble and fall with irrelevancies.
So in words, the Euclid Infinitude of Primes proof, Indirect in
short- form goes like this:
1) Definition of prime
2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
finite with P_k the last and final prime
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) W+1 is necessarily prime
5) contradiction to P_k as the last and largest prime
6) set of primes is infinite.
INDIRECT (contradiction) Method, Long-form; Infinitude of Primes
Proof
and the numbering is different to show the reductio ad absurdum
structure
as given by Thomason and Fitch in Symbolic Logic book.
(1) Definition of prime as a positive integer divisible
only by itself and 1.
(2) The prime numbers are the numbers 2,3,5,7,11, ..,pn,... of set S
Reason: definition of primes
(3.0) Suppose finite, then 2,3,5, ..,p_n is the complete series set
with p_n the largest prime Reason: this is the supposition step
(3.1) Set S are the only primes that exist Reason: from step (3.0)
(3.2) Form W+1 = (2x3x5x, ..,xpn) + 1. Reason: can always operate and
form a new number
(3.3) Divide W+1 successively by each prime of
2,3,5,7,11,..pn and they all leave a remainder of 1.
Reason: unique prime factorization theorem
(3.4) W+1 is necessarily prime. Reason: definition of prime, step
(1).
(3.5) Contradiction Reason: pn was supposed the largest prime yet we
constructed a new prime, W+1, larger than pn
(3.6) Reverse supposition step. Reason (3.5) coupled with (3.0)
(4) Set of primes are infinite Reason: steps (1) through (3.6)
The reason the mathematics community from Euclid to 1991 could never
do a proof
of the Infinitude of Twin Primes, has two reasons-- (a) that community
could never
do a valid Euclid Infinitude of Regular Primes via reductio ad
absurdum, and that the
Twin Primes proof has only one proof method.
If the Twin Primes proof had several methods such as fetching a
Topology proof or fetching
a Analysis proof or a Series proof or some form of geometry proof,
then the stained and marred
Euclid IP indirect would have been continued to be ignored.
But luckily, in 1991 someone noticed that there was a flaw in the
indirect Euclid IP proof. The
flaw is that once the Euclid number is formed "multiply the lot and
add 1" it is immediately a
new prime and necessarily a new prime under the constraints of the
assumption step.
The often cited example of 1+2x3x5x7x11x13 = 59x509 was only an
irrelevant distraction and only kept Euclid's IP indirect marred in
illogic and unable to do Twin Primes proof.
Once it is recognized and seen that in the Indirect, what you get is a
necessarily two new primes of W+1 and W-1, allows for a quick and easy
proof of the Infinitude of Twin Primes.
Archimedes Plutonium
http://www.iw.net/~a_plutonium/
whole entire Universe is just one big atom
where dots of the electron-dot-cloud are galaxies
Archimedes Plutonium
of
the Euclid Infinitude of Primes proof via indirect (reductio ad
absurdum, or sometimes
called proof by contradiction)
Here I want to show you the proof of Euclid Infinitude of Primes
direct method, both
short-form and long-form.
And Euclid's IP, Direct or constructive in short-form goes like this:
1) Definition of prime
2) Given any finite set of primes
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) Either W+1 is prime or we conduct a prime factor search
5) this new prime increases the set cardinality by one more prime
6) since this operation of increasing set cardinality occurs for
any
given finite set we start with, means the primes are infinite set.
DIRECT Method (constructive method), long-form; Infinitude of Primes
Proof
(1) Definition of prime as a positive integer divisible
only by itself and 1.
(2) Statement: Given any finite collection of primes
2,3,5,7,11, ..,p_n possessing a cardinality n Reason: given
(3) Statement: we find another prime by considering W+1 =(2x3x...xpn)
+1 Reason: can always operate on given numbers
(4) Statement: Either W+1 itself is a prime Reason: Unique Prime
Factorization theorem
(5) Statement: Or else it has a prime factor not equal to any of the
2,3,...,pn
Reason: Unique Prime Factorization theorem
(6) Statement: If W+1 is not prime, we find that prime factor Reason:
We take the square root of W+1 and we do a prime search through all
the primes from 2 to
square-root of W+1 until we find that prime factor which
evenly divides W+1
(7) Statement: Thus the cardinality of every finite set can be
increased. Reason: from steps (3) through (6)
(8) Statement: Since all/any finite cardinality set can be increased
by one more prime, therefore the set of primes is an infinite set.
Reason: going from the existential logical quantifier to the
universal
quantification
Alright, there you have it. Both proofs of Euclid Infinitude of
Primes, one the constructive or direct method and the other the
contradiction or indirect method.
The one major difference between the two proof methods is the step in
which you
construct the Euclid Number, W+1, "multiply the lot and add 1". That
is the step that
differentiates the two methods. In the contradiction method W+1 has to
be necessarily
a new prime. In the contructive method W+1 can be prime or has to have
a prime factor search.
So the flaw of logic of all Euclid Infinitude of Primes proof before
1991, is that everyone
mixed up those two methods. They applied the example of
1+2x3x5x7x11x13 = 59x509 to the
Contradiction method when that is irrelevant. That example of
1+2x3x5x7x11x13 = 59x509
is applicable only to the Constructivist method where a prime factor
is searched for.
In the Contradiction method, the moment W+1 is formed, we cannot look
for a prime factor since the supposition assumption was that the
primes were finite. We are under the laws of
Logical form. And thus, in the Contradiction method our only hope of a
new prime to beget a
contradiction is W+1 itself. And it is necessarily prime because of
the assumption supposition.
Now most readers have a difficult time of believing me as to what I
have said above. So here is a post to sci.math around 1994 where a
different gentleman speaks about the logical necessity that W+1 or
Euclid's number must necessarily be prime.
Karl Heuer gives a correct Euclid IP, indirect method
Sun, 20FEB1994, 21:05:13 GMT sci.math
INCONSISTENT PEANO AXIOMS AND MATH PROFESSORS
Lines: 36
Sender: k...@spdcc.com (Karl Heuer)
k...@ursa-major.spdcc.com (Karl Heuer) writes:
In article (5JChA8g2...@jojo.escape.de>
det...@jojo.escape.de (Detlef Bosau) writes:
>Ludwig.Pluton...@dartmouth.edu meinte am 18.02.94
>>det...@jojo.escape.de (Detlef Bosau) writes:
>>>Wrong. Your two numbers are not necessarily prime
>>NO, YOU ARE WRONG. Those numbers are necessarily prime, due to
>>UPFAT, all the primes that exist in the finite set leave a remainder
>>of 1.
>I'll give you a lesson of elementary arithmetics. . .
I really shouldn't bother to get involved in this discussion again,
but
Ludwig is right. In logical terms, his key statement is "if P is a
finite set containing all the primes, then prod(P)+1 is prime." This
is
a true statement.
Let's step through your alleged counterexample:
>consider your set of primes to be: {2,3,5,7,11,13}, as I assert 13 to be
>the largest prime. [. . .] Now, you made the assertion, that
> > > > (2x3x5x11x13) + 1 [=30031] must be prime.
Yes, it's true that if 13 is the largest prime, then 30031 is prime.
Do
you disagree with that assertion?
>As you stated before, there exists an unique prime decomposition of
>30031. This is 59x509. It could be easily shown, that 59 and 509
>both are prime.
If 13 is the largest prime, then 59x509 is not a factorization of
30031.
--- end quoting Karl Heuer's post of 1994 ---
And now, to contrast the clarity and true valid indirect of Karl Heuer
here is an example of a invalid, messy, and error filled take on
Euclid's Infinitude of Primes as listed in Wikipedia tonight:
--- quoting a paragraph of Wikipedia on Euclid's Infinitude of Primes
proof ---
The proof is sometimes phrased in a way that falsely leads some
readers to think that P + 1 must itself be prime, and think that
Euclid's proof says the prime product plus 1 is always prime. This
confusion arises when the proof is presented as a proof by
contradiction and P is assumed to be the product of the members of a
finite set containing all primes. Then it is asserted that if P + 1 is
not divisible by any members of that set, then it is not divisible by
any primes and "is therefore itself prime" (quoting G. H. Hardy[11]).
This sometimes leads readers to conclude mistakenly that if P is the
product of the first n primes then P + 1 is prime. That conclusion
relies on a hypothesis later proved false, and so cannot be considered
proved. The smallest counterexample with composite P + 1 is
2 × 3 × 5 × 7 × 11 × 13 + 1 = 30,031 = 59 × 509 (both primes).
--- end quoting Wikipedia ---
Can you see how muddle headed and wrong that author was? Can you see
that the Wikipedia
author used the irrelevant example of 2 × 3 × 5 × 7 × 11 × 13 + 1 =
30,031 = 59 × 509
when that example is only applicable to the constructive or direct
method.
When you do the Euclid Infinitude of Primes contradiction method, you
can never point to that irrelevant example of 2 × 3 × 5 × 7 × 11 × 13
+ 1 = 30,031 = 59 × 509 because in the
Indirect method the only primes existing were 2,3,5,7,11,13 and that
means 30,031 is
necessarily prime, contradiction and primes are infinite.
Can you see that when you mix methods, that you fail to have a valid
proof?
And this is the reason that the Infinitude of Twin Primes could never
been proven is
because there is only one method to gain that proof and it happens to
be the Indirect method. But since the Infinitude of Regular Primes had
never been given a valid Indirect
Proof before, so then the Twin Primes could never be proven.
Once you recognize that the past mathematicians never were able to
give a valid Euclid
Infinitude of Regular Primes by contradiction method (they were hung
up on an irrelevant
example), then you realize that never was there going to be a Twin
Primes proof.
Archimedes Plutonium
from
the valid Indirect method proof of Euclid's Infinitude of Regular
Primes. Remember
that the valid indirect proof had W+1 as necessarily prime due to that
hypothetical
assumption step. What then is added is the fact that W-1 would also be
necessarily
prime. So we truly have in Euclid's Infinitude of Regular Primes that
both W+1
and W-1 are necessarily new primes in the proof by contradiction.
But there is one slight problem in the Twin Primes proof, is that of
the interwoven
regular primes and whether the Twin Primes are just finite with the
last two primes
on the end. That problem is overcome by the recursion application.
Now I suspect most people are going to have a frightening difficult
time of evaluating this proof,
and that is to be expected, since most people could never do a valid
Indirect of Regular
Primes without messing and mixing it up with direct method. Here the
complexity of thought is increased exponentially, having to juggle the
regular primes and twin primes all at once.
I am always wanting improvement in my proofs and as time goes by,
hopefully I can find
better ways of explaining or increasing the clarity. A mathematical
proof is in many ways
a reflection of a person's maturing through age. So that as one gets
older, they think back to a situation and analyze or summarize that
situation better due to their increased understanding. Same thing for
a mathematics proof is that this is the best I can offer
at the moment but with more years later, increase the clarity.
Short Form Proof of Infinitude of Twin Primes
(1) definition of prime
(2) hypothetical assumption: suppose the set of all primes including
twin-
primes is finite with the last two and largest primes as twin-primes
and this sequence list is 2,3, 5, 7, 11, . . , p_n, p_n+2
(3) Multiply the lot and add 1 and subtract 1 yielding W-1 and W+1
(4) both W-1 and W+1 are necessarily new primes and twin primes from
(1) and the fact
that successive division by all the primes that exist in (2) leave a
remainder
(5) now form a new sequence of 2, 3, 5, 7, 11, . . , p_n, p_n+2, W-1,
W +1
(6) Multiply this lot and add and subtract 1 yielding Z-1 and Z+1
(7) continue this recursive multiply the lot and yielding two new
numbers
(8) Contradiction to (2) in that these successive new numbers are
larger than previous and are twin primes by the definition of prime
and the fact that all these new numbers
formed leave a remainder upon division
(9) reverse (2) set of Twin Primes is Infinite
Veky
> twin-
> primes is finite with the last two and largest primes as twin-primes
> and this sequence list is 2,3, 5, 7, 11, . . , p_n, p_n+2
> (9) reverse (2) set of Twin Primes is Infinite
No.
Negation of (2) is: the set of all primes (that contains all the twin primes) is infinite. Nowhere do you get that the set of all twin primes is infinite, just that it is contained in the infinite set.
Archimedes Plutonium
No, in the Euclid regular primes proof we need only one new prime for
the contradiction.
Here I build a tower of new twin primes.
I could have built that tower for Regular primes alone, but why bother
since it needed just
one new prime for the contradiction.
Here I save the contradiction up, as I build the tower of new primes.
I started with {2,3,5,7,11, . . , p_n, p_n+2}
then I added on W+1 and W-1 and add those to the original
{2,3,5,7,11, . . , p_n, p_n+2, W-1, W+1}
then I used those to generate more twin primes. I generated Z-1 and Z
+1 and recursively add that into the new batch to generate more twin
primes.
So you see, in Euclid's proof of Regular Primes I could also do this
recursive generator, but
why bother since one new prime already fulfills the infinite set
needed.
For Twin Primes, if we question whether twin primes is finite set with
that last twin primes sticking
on the end of the assumption-step, then the method via recursion
allows us to build like a tower to as far as we
want to go for twin primes.
Archimedes Plutonium
Alright Veky has caused me to reinforce the proof explanation by also
showing a different version of the
proof of Infinitude of Regular Primes.
Euclid Infinitude of Primes proof, Indirect , normal version
1) Definition of prime
2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is finite
with P_k the last and final prime
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) W+1 is necessarily prime
5) contradiction to P_k as the last and largest prime
6) set of primes is infinite.
Euclid Infinitude of Primes proof, Indirect , fancier version
1) Definition of prime
2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is finite
with P_k the last and final prime
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) W+1 is necessarily prime
5) Include W+1 into prior set of { 2,3,5,7,.. , P_k , W+1 }
6) Multiply the lot and add 1 yielding Y+1 and recursively include it
to the prior set
7) Multiply the lot and add 1 yielding Z+1
8) contradiction to P_k as the last and largest prime where we
recursively generated
new primes of W+1, and Y+1 and Z+1 ad infinitum
9) set of primes is infinite.
So thanks to Veky, I have thus made my proof of infinitude of twin
primes that much clearer.
So that when I present the proof of infinitude of twin primes, I
should introduce it with both versions
of the proof of infinitude of regular primes, indirect method.
Short Form Proof of Infinitude of Twin Primes, indirect method
(1) definition of prime
(2) hypothetical assumption: suppose the set of all primes including
twin-
primes is finite with the last two and largest primes as twin-
primes and this sequence list is 2,3, 5, 7, 11, . . , p_n, p_n+2
(3) Multiply the lot and add 1 and subtract 1 yielding W-1 and W+1
(4) both W-1 and W+1 are necessarily new primes and twin primes from
(1) and the fact
that successive division by all the primes that
exist in (2) leave a
remainder
(5) now form a new sequence of 2, 3, 5, 7, 11, . . , p_n, p_n+2, W-1,
W +1
(6) Multiply this lot and add and subtract 1 yielding Z-1 and Z+1
(7) continue this recursive multiply the lot and yielding two new
numbers
(8) Contradiction to (2) in that these successive new numbers are
larger than previous and are twin primes by the definition of
prime and the fact that all these new numbers formed leave a remainder
upon division
(9) reverse (2) set of Twin Primes is Infinite
Archimedes Plutonium
Veky
> twin- primes is finite with the last two and largest primes as twin-
> primes and this sequence list is 2,3, 5, 7, 11, . . , p_n, p_n+2
> (9) reverse (2) set of Twin Primes is Infinite
You still have exactly the same problem.
In (2), you hypothesised that set of all primes, including the set of twin primes, is finite.
In (9), (since you have reached a contradiction) you can thus conclude negation of it. That negation ("reverse" as you call it) is _not_ that set of twin primes is infinite. The negation of (2), as I said already, is that set of all primes, including the set of twin primes, is infinite. And it doesn't imply that set of twin primes is infinite.
Please note, everything else in your proof is "correct" (quotes because it is in contradictional universe), and you don't need to clarify it. However, you need to understand what is the negation of the claim "set A, containing set B, is finite". It does not imply that set B is infinite.
Archimedes Plutonium
Alright, thanks again Veky. I forget to include the "logical
conjunction" in that (2) statement.
Once I include it by saying both regular-primes are supposed finite
and twin primes are
supposed finite, the discharge of the hypothetical assumption converts
both the regular
primes and twin primes into the infinite set category.
Short Form Proof of Infinitude of Twin Primes
(1) definition of prime
(2) hypothetical assumption: suppose the set of regular primes and
(logical conjunction)
twin- primes is finite with the last two and
largest primes as twin-primes
and this sequence list is 2,3, 5, 7,
11, . . , p_n, p_n+2
(3) Multiply the lot and add 1 and subtract 1 yielding W-1 and W+1
(4) both W-1 and W+1 are necessarily new primes and twin primes from
(1) and the fact
that successive division by all the primes that
exist in (2) leave a
remainder
(5) now form a new sequence of 2, 3, 5, 7, 11, . . , p_n, p_n+2, W-1,
W +1
(6) Multiply this lot and add and subtract 1 yielding Z-1 and Z+1
(7) continue this recursive multiply the lot and yielding two new
numbers
(8) Contradiction to (2) in that these successive new numbers are
larger than previous and are twin primes by the definition of prime
and the fact that all these new numbers formed leave a remainder upon
division
(9) reverse (2) set of Twin Primes is Infinite
Archimedes Plutonium
David R Tribble
>> (2) hypothetical assumption: suppose the set of all primes including
>> twin- primes is finite with the last two and largest primes as twin-
>> primes and this sequence list is 2,3, 5, 7, 11, . . , p_n, p_n+2
>> (9) reverse (2) set of Twin Primes is Infinite
>
Veky wrote:
>> You still have exactly the same problem.
>>
>> In (2), you hypothesised that set of all primes, including
>> the set of twin primes, is finite.
>>
>> In (9), (since you have reached a contradiction) you can
>> thus conclude negation of it. That negation ("reverse" as you
>> call it) is _not_ that set of twin primes is infinite.
>> The negation of (2), as I said already, is that set of all primes,
>> including the set of twin primes, is infinite. And it doesn't
>> imply that set of twin primes is infinite.
>>
>> Please note, everything else in your proof is "correct"
>> (quotes because it is in contradictional universe), and you
>> don't need to clarify it. However, you need to understand
>> what is the negation of the claim "set A, containing set B, is finite".
>> It does not imply that set B is infinite.
>
Archimedes Plutonium wrote:
> Alright, thanks again Veky. I forget to include the "logical
> conjunction" in that (2) statement.
> Once I include it by saying both regular-primes are supposed finite
> and twin primes are
> supposed finite, the discharge of the hypothetical assumption converts
> both the regular
> primes and twin primes into the infinite set category.
The negation of "Set A is finite and set B is finite" is not
"Set A is infinite and set B is infinite". The negation is
"not (A is finite and B is finite)" which is
"(A is infinite) or (B is infinite)". Read up on DeMorgan's
Rule for details.
In this particular case, since B is a subset of A, the
clause "A is infinite" is true, while :"B is finite" remains
either true or false. Which means you're back to square
one, having to provie that set B is not finite.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
> On Jan 25, 11:39 pm, Veky <ved...@gmail.com> wrote:
>
> > > (2) hypothetical assumption: suppose the set of all primes including
> > > twin- primes is finite with the last two and largest primes as twin-
> > > primes and this sequence list is 2,3, 5, 7, 11, . . , p_n, p_n+2
> > > (9) reverse (2) set of Twin Primes is Infinite
>
> > You still have exactly the same problem.
>
Alright, Veky, I really do not need the "logical conjunction" in the
hypothetical
assumption.
I can work, alone, with the fact of suppose Primes are finite. It is
the tower
of twin primes that I build within that hypothetical assumption that
releases the
assumption. It is also known that an infinite set can have an infinite
subset.
> > In (2), you hypothesised that set of all primes, including the set of twin primes, is finite.
>
> > In (9), (since you have reached a contradiction) you can thus conclude negation of it. That negation ("reverse" as you call it) is _not_ that set of twin primes is infinite. The negation of (2), as I said already, is that set of all primes, including the set of twin primes, is infinite. And it doesn't imply that set of twin primes is infinite.
>
> > Please note, everything else in your proof is "correct" (quotes because it is in contradictional universe), and you don't need to clarify it. However, you need to understand what is the negation of the claim "set A, containing set B, is finite". It does not imply that set B is infinite.
>
> Alright, thanks again Veky. I forget to include the "logical
> conjunction" in that (2) statement.
> Once I include it by saying both regular-primes are supposed finite
> and twin primes are
> supposed finite, the discharge of the hypothetical assumption converts
> both the regular
> primes and twin primes into the infinite set category.
>
I do not need that. That was overkill. I can use the fact that I can
build ad-infinitum
twin primes from the singular hypothetical assumption that regular
primes are finite.
Short Form Proof of Infinitude of Twin Primes
(1) definition of prime
(2) hypothetical assumption: suppose the set of regular primes is
finite with the twin-primes as a subset embedded inside the regular
primes and also finite because the regular primes are finite in this
hypothetical assumption, and
this sequence list is {2,3, 5,
7,11, . . , p_k}, where p_k is the largest finite prime.
(3) Multiply the lot and add 1 and subtract 1 yielding W-1 and W+1
(4) both W-1 and W+1 are necessarily new primes and twin primes from
(1) and the fact
that successive division by all the primes that
exist in (2) leave a
remainder
(5) now we recursively continue to multiply the lot and add 1,subtract
1 by adjoining the previous.
(6) now form a new sequence of 2, 3, 5, 7, 11, . . , p_k, W-1,W +1
(7) Multiply this lot and add and subtract 1 yielding Z-1 and Z+1
(8) continue this recursive multiply the lot and yielding two new
numbers
(9) This recursive adjoining yields pairs of numbers ad infinitum
(10) Contradiction to (2) in that these successive new pairs of
numbers (tower of built pairs of numbers) are
larger than p_K the supposed largest finite prime, and are twin primes
by the definition of prime
and the fact that all these new numbers formed are ad infinitum and
they leave a remainder upon
division, reverse (2) set of primes having Twin Primes as subset is
infinite. Since twin primes are a subset of primes and since infinite
sets have infinite subsets, regular-primes are also infinite.
>
> Archimedes Plutoniumhttp://www.iw.net/~a_plutonium/
Archimedes Plutonium
I am still not happy with the above Veky, so let me try one more
attempt.
Euclid Infinitude of Primes proof, Indirect , normal version
1) Definition of prime
2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
finite
with P_k the last and final prime
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) W+1 is necessarily prime
5) contradiction to P_k as the last and largest prime
6) set of primes is infinite.
That is the normal version without building a Tower of Infinity.
But let me see how a Tower of Infinity makes the above even better.
Euclid Infinitude of Primes proof, Indirect , tower- version
1) Definition of prime
2) Hypothetical assumption, suppose set of primes 2,3,5,7,.. is
finite
with P_k the last and final prime
3) Multiply the lot and add 1 (Euclid's number) which I call W+1
4) W+1 is necessarily prime
5) Include W+1 into prior set of { 2,3,5,7,.. , P_k , W+1 }
6) Multiply the lot and add 1 yielding Y+1 and recursively include
it
to the prior set
7) Multiply the lot and add 1 yielding Z+1
8) contradiction to P_k as the last and largest prime where we
recursively generated
new primes of W+1, and Y+1 and Z+1 ad
infinitum
9) set of primes is infinite.
Alright, not only did we reverse the assumption step, but we actually
built a Tower of
Infinite Primes, for which we note in the reverting of the assumption
step. This should
be applicable to the infinitude of Twin Primes that once we have built
that tower ad infinitum of Twin Primes, that the last step not only
recognizes the infinitude of regular primes but the twin primes as
well.
Short Form Proof of Infinitude of Twin Primes, indirect method
Short Form Proof of Infinitude of Twin Primes
(1) definition of prime
(2) hypothetical assumption: suppose the set of regular primes is
finite with the twin-primes as a subset embedded inside the regular
primes and also finite because the regular primes are finite in this
hypothetical assumption, and
this sequence list is {2,3, 5,
7,11, . . , p_k}, where p_k is the largest finite prime.
(3) Multiply the lot and add 1 and subtract 1 yielding W-1 and W+1
(4) both W-1 and W+1 are necessarily new primes and twin primes from
(1) and the fact
that successive division by all the primes that
exist in (2) leave a
remainder
(5) now we recursively continue to multiply the lot and add
1,subtract
1 by adjoining the previous.
(6) now form a new sequence of 2, 3, 5, 7, 11, . . , p_k, W-1,W +1
(7) Multiply this lot and add and subtract 1 yielding Z-1 and Z+1
(8) continue this recursive multiply the lot and yielding two new
numbers
(9) This recursive adjoining yields pairs of numbers, twin primes, are
not evenly divisible, and are ad infinitum, so reverse the assumption.
(10) Twin primes are infinite
What I have done is weaved through the fact that a supposition of the
regular primes as a finite set, fails to stop a building tower of Twin
Primes that goes to infinity. There maybe, still, some poor grammar or
omissions of minor details, but the overall logical form
and gist is there.
Again, I do not need "logical conjunction" and it only adds excess
baggage.
Now perhaps by deletion of some primes, I can prove the infinitude of
quad primes separated
not by 2 as in twin primes but separated by a metric of 4, and then
prove hex primes separated by 6 metric, ad infinitum. Using the same
method as above only that we trim away
some starting primes such as 2, and for hex primes strip away 2, and
3.
Archimedes Plutonium
Archimedes Plutonium
infinitude of
N+2 primes = Twin primes, N+4 primes = quad primes, N+6 primes = hex
primes.
Anyway, since the valid Euclid Infinitude of Primes proof has Euclid's
number W+1 and W-1
as necessarily new primes, we can work that fact for proofs of twin
primes, quad primes,
hex primes and all other even numbered spaced primes.
In July of 2010 I sent out some posts in proving infinitude of N+4
primes Quad Primes.
But am anxious to try the Tower Ad Infinitum approach on Quad, Hex etc
etc primes.
sci.math, sci.logic
Jul 12, 1:12 pm
Date: Jul 12, 2010 2:12 PM
Author: plutonium....@gmail.com
Subject: square-root eliminator works for all even separated primes,
for
example N+4 primes
Archimedes Plutonium wrote:
(snipped)
> There maybe a sticking point about this proof procedure for large even
> numbered
> prime pairs such as say p_n and p_n+1000, that we may have to have
> some
> more qualifiers to impose the Square Root Patch, due to the large
> spread between
> the n and even numbered n_
Let me write a list of the first few primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, . .
The first twin primes is 3, 5
The first quad primes (n+4) is 3, 7
The first (n + 6) primes is 5, 11
The first (n +8) primes is 3, 11
etc etc
And so it matters not, how large the N+2k becomes, because the
square-
root
patch on the last prime in the successive prime list captures all
the
primes
of that succession, that finite succession, for example, doing the
N +4 primes proof would go like this:
Proof of the Infinitude of N+4 Primes:
(1) Definition of prime
(2) Hypothetical Assumption, suppose set of all N+4 Primes is
finite
with
3,7 being the last two n+4 primes of the sequence set S = 3, 5, 7
where
7 is the last and largest of the N+4 primes. Note we can either use
2 in the S list or delete 2, depending on convenience.
(3) Multiply the lot and add 2 and subtract 2, yielding W+2 =
(3x5x7)
+2 = 107 and
W-2 = (3x5x7) -2 = 103
(4) Take the square root of W+2 and I get 10.3.... I do not need
square root of W-2, and there cannot be any
regular primes
for consideration of being a prime factor of this W+2 that is
greater
than 10
(5) Successively divide all the primes in the sequence S into W+2
and
W-2 and
they all leave a remainder
(6) The two new numbers W+2 and W-2 are necessarily two new N+4
primes
(7) Contradiction to 3 and 7 with 7 the largest and last N + 4
primes
(8) Set of N+4 Primes are infinite.
Archimedes Plutonium
infinitude
of them is intimately connected with the proof of Twin Primes. So that
a valid
Twin Primes proof should also afford to do valid proofs of Quad, Hex,
etc etc primes.
The reason Twin Primes infinitude was held back in history from ever
having a proof
was the fact that no-one could do a valid infinitude of regular primes
indirect method.
Once that is removed and a valid indirect regular primes is given
where W+1 and W-1
are necessarily new primes, opens the door to proofs of Twin and Quad
and Hex and all
even numbered spaced primes.
So I have to see, now, if that Tower ad infinitum gives easy proofs of
quad and hex and
all other evenly spaced primes. Apparently in 2010, I used a square
root technique, but I think that was excess baggage that I did not
need and can do quad and hex by that simple
Tower ad infinitum technique used in Twin Primes.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
> I forgotten the name of this conjecture. It says something about the
> infinitude of
> N+2 primes = Twin primes, N+4 primes = quad primes, N+6 primes = hex
> primes.
>
> Anyway, since the valid Euclid Infinitude of Primes proof has Euclid's
> number W+1 and W-1
> as necessarily new primes, we can work that fact for proofs of twin
> primes, quad primes,
> hex primes and all other even numbered spaced primes.
>
> In July of 2010 I sent out some posts in proving infinitude of N+4
> primes Quad Primes.
>
> But am anxious to try the Tower Ad Infinitum approach on Quad, Hex etc
> etc primes.
>
> sci.math, sci.logic
> Jul 12, 1:12 pm
> Date: Jul 12, 2010 2:12 PM
Alright, I got off to a rocky, and stumbling start there with the
proof of the
Infinitude of Twin primes and questions asked by Veky. Those were good
questions
for they showed errors in my steps, but those errors are easily
amended.
The trouble or stumbling points was how to interweave a proof of
infinitude of Twin, or
Quad or Hex etc etc primes along with the Regular Primes? How to make
that interwoven
proof flow without any logical stumbles.
The thing about the proof of Infinitude of Twin Primes is that it
relied on a valid Indirect Method of Infinitude of Regular Primes
where W+1 and W-1 were necessarily two
new primes in indirect.
So the feature that W+1 and W-1 are necessarily new primes is all that
is needed to render
a proof of the Infinitude of Twin Primes, and quad and hex etc etc.
No-one in math history ever produced a valid indirect proof of
infinitude of regular primes but once that is produced, it allows the
proof of infinitude of twin, quad, hex etc primes.
And it does so that no fancy extra has to be included in the proof.
That the proof of the Infinitude of Twin Primes is as easy and
streamlined is the proof of Regular Primes.
Indirect method proof of the Infinitude of Regular Primes:
(1) Definition of prime
(2) Hypothetically suppose p_k is the last final prime
(3) form the sequence 2,3,.., p_k
(4) Multiply the lot and add, subtract 1 producing W-1 and W+1
(5) W-1 and W+1 are necessarily two new primes
(6) contradiction to (2) and regular primes are infinite
Indirect method proof of the Infinitude of Twin Primes:
(1) Definition of prime
(2) Hypothetically suppose p_n and p_n+2 are the last and final twin
primes
(3) form the sequence 2,3,.., p_n, p_n+2
(4) Multiply the lot and add, subtract 1 producing W-1 and W+1
(5) W-1 and W+1 are necessarily two new twin-primes
(6) contradiction to (2) and twin primes are infinite
Indirect method proof of the Infinitude of Quad Primes:
(1) Definition of prime
(2) Hypothetically suppose p_n and p_n+4 are the last and final quad
primes
(3) form the sequence 3,.., p_n, p_n+4
(4) Multiply the lot and add, subtract 2 producing W-2 and W+2
(5) W-2 and W+2 are necessarily two new quad-primes
(6) contradiction to (2) and quad primes are infinite
Indirect method proof of the Infinitude of Hex Primes:
(1) Definition of prime
(2) Hypothetically suppose p_n and p_n+6 are the last and final hex
primes
(3) form the sequence 3,.., p_n, p_n+6
(4) Multiply the lot and add, subtract 3 producing W-3 and W+3
(5) W-3 and W+3 are necessarily two new hex-primes
(6) contradiction to (2) and hex primes are infinite
The above proves the infinitude of all evenly spaced metric of primes.
What makes the above work is that the Indirect Method, masks or cloaks
the specific
number of W-1 and W+1 and reaches for and grabs the general W-1 and W
+1 as having
to be **necessarily a prime pair**. If this were Direct Method, we
have to conduct
prime factor searches and prime factor inspections all over the place,
but because it is Indirect, we simply know that the method
reaches in and grabs a pair of primes that contradicts the
hypothetical assumption.
Again, repeating, in Old-Math, no-one was able to put together a valid
indirect method
proof of the Infinitude of Regular Primes. This lack of a valid proof
where W-1 and W+1
are necessarily new primes, thwarted the proof of the Infinitude of
Twin Primes. Once
a valid Infinitude of Regular Primes, indirect method is given, it
immediately is turned into a proof of the infinitude of twin primes
and all other evenly spaced primes.
All that is needed is the guarantee that the "multiply the lot and add
subtract" are
necessarily two new primes, and that is what was missing in Old-Math
as they gazed upon
twin primes.
Now I can insert number examples in the proof above such as for Quad
primes.
So we insert the sequence {3,5,7} Now we multiply the lot and add
subtract 2, yielding
105-2 and 105+2 for that of 103 and 107. Now maybe or maybe not that
103 and 107 are two
quad primes. Regardless, of whether a specific example is or is not a
quad primes, the
important message is that in the Indirect method, they are always two
new quad primes
due to the hypothetical assumption. For it is the *** logical format
*** of the proof that
forces them to be two new quad primes not listed on that finite list.
Archimedes Plutonium
Looking through my old posts it was called the Polignac Conjecture.
Here is what Wikipedia says about it:
--- quoting Wikipedia ---
Polignac's conjecture from 1849 states that for every even natural
number k, there are infinitely many prime pairs p and p′ such that p −
p′ = k. The case k = 2 is the twin prime conjecture.
--- end quoting Wikipedia ---
Alright, since we have the liberty of producing two new primes in the
Euclid Infinitude of Primes proof, Indirect Method,
in which W-1 and W+1 are guaranteed to be two new primes, we can
thence prove not only the Twin Primes conjecture
but easily the Polignac Conjecture. Following the same proof pattern
above here is a proof of the Polignac Conjecture.
Indirect method proof of the Polignac Conjecture
(1) Definition of prime
(2) Hypothetically suppose p_n and p_n+2k are the last and final
specific polignac primes
(3) form the sequence 3,.., p_n, p_n+2k
(4) Multiply the lot and add, subtract k producing W-k and W+k
(5) W-k and W+k are necessarily two new polignac-primes
(6) contradiction to (2) and polignac primes are infinite
Now why is all this possible when Twin Primes, let alone all evenly
spaced primes are infinite sets
when Twin Primes is as old of a conjecture as Ancient Greek times? How
can all of these conjectures
be proven so quickly and easily?
The answer is quite simple and easy. In that never before has a
mathematician been able to give a valid
proof, indirect method of the infinitude of regular primes. Everyone,
before, were mixing up direct with indirect,
unable to realize that the example of 1+(2x3x5x7x11x13) = 59x509
applies only to the Direct Method and not
the Indirect Method. Everyone that did a Euclid Infinitude of Primes,
before, passed off the Euclid IP proof as
if the direct method was exactly identical to the indirect method
because they felt that the example applied to
both methods. They were logically handicapped in not realizing that
the Indirect method forced Euclid's Number
to be necessarily a pair of new primes of W-1 and W+1. So that
everyone who did Euclid's Infinitude of Primes proof
had not the benefit of a valid Indirect method where you can use the
fact that W-1 and W+1 are necessarily two
new primes.
Using the fact that W-1 and W+1 are two new necessary primes in the
Infinitude of Primes Proof easily allows the
proof of Twin primes and Polignac Primes.
Also, it must be commented that the proof of Infinitude of Twin Primes
has only one proof method-- indirect number theory proof. For if it
allowed a geometry or topology or Series or analysis proof, then a
Twin Primes Infinitude proof
would have been discovered and the invalid Indirect proofs ignored.
But since Twin Primes has only one proof method,
we have to set straight the invalid past indirect proofs, correct
them, and then we can do the Twin Primes proof.
Archimedes Plutonium
infinitude of twin primes
although working well on the Polignac Conjecture could then be
transfered to prove the infinitude
of Mersenne primes and thus the infinitude of perfect numbers. As it
turned out, that was not true
for the transfer could not be made to Mersenne Primes, because the
method does not guarantee
the W+1 or W-1 has the same form as (2^p) - 1.
Now the reason I want to post this is to show that a true method is
limited in what it can prove. If
the method could prove the infinitude of any sort of primes offered,
then we would suspect the method
overall is false. A true method in mathematics is very limited.
sci.math, sci.logic
Jul 13, 1:49 pm
Date: Jul 13, 2010 2:49 PM
Author: plutonium....@gmail.com
Subject: Proof of the Infinitude of Perfect Numbers and infinitude of
Mersenne
primes
Archimedes Plutonium wrote:
(snipping)
Now we have a proof of the Infinitude of Perfect Numbers and Mersenne
primes.
I leave it to the reader to look up what they mean. I am just
showing
what the proof is
and expect the reader to know what the problems were. But I do make
note of the history.
This is perhaps the oldest unsolved mathematics problem, along with
1
being the only
odd perfect number. The reason that I am able to prove it, is
because
of a tiny small mistake
and misunderstanding in the Indirect Proof method. In that method,
there is a step where
Euclid's Number under view is "necessarily a new prime within the
Indirect Logic structure"
This allows for the proof.
Proof of Infinitude of Perfect Numbers and Infinitude of Mersenne
Primes:
(1) definition of prime
(2) hypothetical assumption: suppose set of all primes is finite
and 2,3,5, 7, 11, . ., ((2^p) - 1) is the complete list of all the
primes with
((2^p) - 1) the last and largest prime.
(3) Form Euclid's numbers of W+1 = (2x3x5x 7x 11x . .x (((2^p) - 1))
+1
and W -1 = (2x3x5x 7x 11x . .x (((2^p) - 1)) -1
(4) Both W+1 and W -1 are necessarily prime because when divided by
all the primes that exist into W+1 and W-1 they leave a remainder
and so they are necessarily prime from (1) and (2)
(5) Contradiction to (2) that W+1 and W-1 are larger primes than
((2^p) - 1).
(6) And W+1 is a prime of form (2^p) + 1, and W -1 is a prime of
form
(2^p) - 1)
Reason: you can place any form
of algebraic prime (x^p) for the last prime in the series so long as
it is -1 or +1 addition
(7) Mersenne primes are an infinite set, hence Perfect numbers are
infinite set.
--- end quoting old post ---
Now the error above is that there is no guarantee that W+1 has the
Mersenne prime form
so that is not a proof.
If my memory is correct, soon after July 2010 I made an alternate
proof of Infinitude
of Mersenne primes using some Mathematical Induction, and I would be
curious if that
turned out to be true or false as a proof.
The point of this post is to show that the method that conquered
infinitude of twin primes and Polignac
conjecture is limited to only that form of primes.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
(snipped)
To do a proper mathematics unsolved proof, you have to go to hell and
back
several times before the proof yields.
Let me not give up so easily as above in prior post. Instead, let me
try again
and maybe I can coax out a form (2^p)-1 for Mersenne primes.
The first few Mersenne primes are these:
3
7
31
127
8191
131071
54524287
Now notice that in the Infinitude of Primes proof to get the Euclid
Number
we multiply the lot and either add or subtract 1. So let me see if I
can get
a Mersenne Prime as a Euclid Number.
(2x3) +1 is indeed the Mersenne prime of 7
(2x3x5) +1 is indeed another Mersenne prime of 31
So indeed, we can go down the line in Euclid Numbers and retrieve on
occasion
a Mersenne Prime
But I need to see if we can retrieve a Mersenne Prime as a Euclid
Number if one
of the primes that compose the Euclid Number is a Mersenne Prime. And
for that
Mersenne prime to stick out on the end of the multiplication string.
So then examining this
(2x3x5x7)+1 and no luck
Now the Mersenne primes get too large too rapidly.
But I am thinking that there is a case example of where the Euclid
number multiplication
string has a Mersenne prime sticking at the end and the W+1 result is
another Mersenne Prime.
A string such as this:
(2x3x5x7x11x13x17x19x23x29x31) +1 and whether it equals another new
Mersenne prime.
If that exists or is true, then a proof that Mersenne primes are
infinite is easily begot
via that Infinitude of primes proof indirect method.
Veky
Now on to more interesting errors:> Indirect method proof of the Infinitude of Twin Primes:
> (1) Definition of prime
> (2) Hypothetically suppose p_n and p_n+2 are the last and final twin
> primes
> (3) form the sequence 2,3,.., p_n, p_n+2
> (4) Multiply the lot and add, subtract 1 producing W-1 and W+1
> (5) W-1 and W+1 are necessarily two new twin-primes
Not necessarily. _If_ they are primes, then they are twin primes. But they don't have to be primes at all. Any of them can have a prime factor bigger than p_n+2. Remember, you just supposed that p_n+2 is the biggest twin prime, not the biggest prime generally.
(There are primes that are not part of any twin prime pair - for example, 23.)
Archimedes Plutonium
Alright, I found a beautiful technique of proving whether a set of
primes is going to be finite or infinite.
Consider the above Mersenne primes discussed and the example:
(2x3x5x7x11x13x17x19x23x29x31) +1 and whether it equals another new
Mersenne prime.
If I can find one example of Mersenne primes where the tip-end
technique is true such that the 31 Mersenne prime
produces a Euclid Number which is also a Mersenne prime then Mersenne
primes are infinite set.
If I cannot find one example of tip-end technique on Euclid's Number,
and then prove that none is possible, I will
have proven Mersenne primes are a finite set.
So the technique, call it tip-end-Euclid's Number is a general
technique that will provide a proof of many classes of
primes of algebraic form as to whether they form a finite set or
infinite set.
For example, are the primes of form (((2^p)^p)^p) -1 are they finite
or infinite? Well here we can easily see that they
become large too fast that when we stick a tip-end prime of that
representative set onto the end of a Euclid Number
"multiply the lot and add or subtract 1" that it is impossible to
equal another prime of that form. So that set of primes is
finite.
So here I found a technique that can easily prove of a prime set of
algebraic form is either finite or infinite, depending on whether a
Euclid Number built from a representative member can produce another
member of that set, and if that
set is impossible to yield another Euclid number prime means it is
finite.
Looking at those Mersenne primes, they build too rapidly that the tip-
end Euclid Number may find a equal prime by
jumping over many of its predecessors.
The Euclid Number with 31 as tip-end prime
(2x3x5x7x11x13x17x19x23x29x31) +1 and whether it equals another new
Mersenne prime.
is impossible to equal 127 or 8191 or 31071 or 524287 but it may equal
some Mersenne prime beyond
524287.
So if I construct a schemata showing that no Mersenne prime as tip-end
of a Euclid Number could possible
equal another Mersenne prime, I will have proven Mersenne primes are
finite.
Now one item I am noticing is that the Mersenne primes after 3 all
seem to end in either a 1 or 7 digit. Is that true?
If that is true, then can I show that no Euclid tip-end Mersenne prime
of its ending 1 and 7 digit is able to reproduce
another new Mersenne tip end prime in a Euclid Number?
Archimedes Plutonium
Now if that is true that all Mersenne primes after 3 end in either a 1
or 7 digit, then
that favors a tip-end-Euclid-Number to find a Mersenne prime as a
Euclid Number. For example
with 7 as tip end we have Euclid Number as (2x3x5x7)+1 = 211 which
ends with a 1 digit, however
211 is not a Mersenne prime. So if we can find one example of where a
tip-end-Euclid-Number
with that tip end as a Mersenne prime equal to another Mersenne prime,
we will have proven
Mersenne primes an infinite set. On the other hand, if we can prove
that no matter what Mersenne prime
is stoked for that tip-end of a Euclid Number since it would never
yield another Mersenne prime,
then we would have proven the opposite that the Mersenne primes are a
finite set.
The chances are in favor that Mersenne primes are infinite.
Archimedes Plutonium
Now there is a major test on the Tip-End-Euclid-Number technique to
tell if a
set of primes is finite or infinite. Sorry that I forgotten the case
example of a
set of primes that looks as though they are heading for infinity, but
in fact ends
up being a finite set afterall. I forgotten the algebraic form of that
set,that looks
as though it was infinite but ends up being finite. I wish I could
remember, and then
plug it into this tip-end technique to see if the tip-end is a
universal type of technique.
Archimedes Plutonium
primes of a form --
are finite or infinite and it turned out that such a algebraic form
ended up finite. So we
need those stark examples of algebraic form where we would think it is
infinite set of primes
but instead runs out finite.
Now let me try the Conjecture of primes of form n^2 +1 which would be
the set
{ 2, 5, 17, 37, 101, . . . }
Is it finite or infinite set?
So let me apply the Tip-End-Euclid's-Number technique
Can I get a prime of form n^2+ 1 when the last tip end of the Euclid's-
Number is a prime
of form n^2 +1
So does (2x3x5) +1 or (-1) yield another prime of form n^2+1 answer is
no
So does (2x3x5x7x11x13x17) +1 or (-1) yield another prime of form n^2
+1 and answer is no.
But these primes get large rapidly and just a few trials are not going
to answer the question.
But from first impressions the primes get big too fast and I doubt
there is ever the case where
the tip end Euclid Number is going to catch another prime of form
n^2+1 resulting in the fact that such a
prime set is finite.
I wish I could remember that old time example of where a algebraic
form of primes is finite and not infinite.
Archimedes Plutonium
Alright, I could say I staged the above concerns about Mersenne Primes
(2^p)-1,
and about n^2+1 primes, let alone Fermat primes of 2^(2^n) +1.
How many times in sci.math have I proposed some new method or
technique to
try to prove a huge lot of prime conjectures of infinity? I recall
more than three times,
none of which passed but failed. If mathematics has all its
definitions in order, we
should not have a backlog of unproven conjectures like this. What the
finite versus
infinite question leaves open to us is the huge amount of unsolved
problems of this
nature, indicating or signalling that the definition of what we think
is infinity is far out
of whack with the reality of infinity.
We should have the situation where we have a Prime Fair for Solutions.
A place where
anyone with a prime conjecture questioning finite versus infinite,
hops into the Fair,
asks the panel of experts whether a given conjecture is finite or
infinite such as Fermat's
primes, or Mersenne Primes or n^2+1 primes, asks the panel and about
an hour or less
later, is given the valid and complete answer. If all the definitions
in math, especially finite
versus infinite were in order, then this Prime Fair is up and running
well and can do that job.
What is different about this edition is that in the 2nd edition I had
not yet solved where the
borderline of Infinity lies and could only attack all these
conjectures of primes whether finite
or infinite in this 3rd edition.
So now, in the proof that 10^603 is the border of Finite versus
Infinite, done via the Circumferencing
the Perimeter, that due to pi having three zeroes in a row at 10^-603
causes a finite perimeter whose
pattern of digits is 215215-- to fail to have a finite circumference
match it. This means that Infinity borderline
exists at 10^603.
This idea that the border of finite ends at 10^603 and infinity starts
there is the main idea in a Prime Fair.
The domain of infinity would be 10^1206 and the range is 10^603, in
order to have completeness Algebra
of add, subtract, multiply and divide. Mathematics ends at 10^603
because Aristotelian Logic ends at
10^603 but we are allowed to reach into 10^1206 to complete the
Algebra.
So getting back to the Prime Fair where the panel of experts is
showing the viewing audience why Regular Primes
and Twin Primes are infinite, because the set of Regular Primes at
10^603 is a set that is 10^600 large in cardinality
and we only have to go to about 10^608 into the 10^1206 territory to
fetch 10^603 cardinality of primes. To fetch the
cardinality of 10^603 Twin Primes we have to go a bit further, but
well under 10^1206 permitted.
So here we are at the Prime Fair and the first proof is blurted out to
the panel of experts: (1) Prove the Polignac Conjecture that all
evenly spaced metric primes are either finite or infinite. Here the
panel of experts tells the
person that some of the evenly spaced primes are infinite while others
are obviously finite. Obviously primes
evenly spaced by 10^603 itself will have only 1 prime and only a few
in the 10^1206 territory so that obviously there
is a Finite cutoff point for evenly spaced primes. We can fetch 10^603
quad primes and hex primes, but once we get to
primes spaced something on the order of increasing of 1000 units apart
before 10^40 is reached, we lose the ability to have an infinite set
of primes in the Polignac conjecture.
Next entry from a group of participants asks for the proof of Mersenne
Primes (2^p)-1. The panel of experts explains
that these sort of primes:
{3 ,
7 ,
31 ,
127 ,
8191 ,
131071 ,
54524287, . .} are measured for a
specific rate of increase and whether that rate of
increase allows that set to be within 10^603 cardinality since we can
reach upwards of 10^1206. So the question is,
is the rate of increase of the existence of Mersenne Primes a rate
that gives us 10^603 such primes if all we can
use are 10^1206 scope of play. Does the Mersenne primes become
stretched apart by more than 1000 at some point that is well before
10^40 is reached? From the above, it appears the Mersenne Primes are
stretching apart by a factor of 100 in just the first seven such
primes, which indicates that not too far away the stretching out
will be a factor of increasing 1000 and increasing beyond a 1000
before 10^40 is reached. This is what happens with the Mersenne
primes, they reach a breaking separation early on in their sequence
that they cannot furnish 10^603 cardinality of primes even though they
can go to 10^1206.
The next group of participants asks for the proof of Fermat Primes and
the panel of experts repeats the Mersenne argument that they have a
breakout separation early on.
The next group of participants asks about the proof of primes of form
n^2+1 and there is an easy proof of this because
we want 10^603 cardinality of primes of that form, yet only allowed to
go to 10^1206 at maximum, yet we are constrained to have only primes
of the squaring of n, and thus the n^2 is 10^603 x 10^603 = 10^1206.
So these primes
are finite.
And the Prime Fair answers all the conjectures of whether finite or
infinite, which was a backlog in mathematics
of a roll-call of over a thousand conjectures wanting to know if a
specific set of primes was finite or infinite.
Whenever a situation like this comes up in the body heart of
mathematics, it is not a situation in which it needs thousands of
clever mathematicians chasing down proofs. It needs just one
mathematician to honestly point to the
idea that Old Math had crumby definitions that needed rectifying and
only after the rectifying of lousy definitions
are all those backlog of conjectures proven in one fell swoop.
Archimedes Plutonium
Alright, we ask for the function that models the rate of separation
increase in order
for that set of primes to be finite and not infinite. To be infinite
means we
can fetch 10^603 primes when the domain is 10^1206. That means there
has to be a
rate of change of separation for the primes that they can come under
the wire of
of 10^1206 and have a set cardinality of 10^603 of primes of that
specific algebraic form,
or that specific sequence of primes.
From what I can discern there is a maximum Polignac evenly spaced
primes that gives the minimum
separation distance per metric distance and still be infinite set.
Obviously primes spaced apart by 10^603 are finite and
primes spaced apart by 2 (twin primes) are infinite since we only have
to go to about 10^615 to fetch
10^603 twin primes. So, somewhere between twin primes and 10^603
spaced primes is a Polignac prime set
that starts to be finite and not infinite. I would characterize that
set of Polignac primes as a minimum
set to be infinite. And I would want to analyze its separation
distance.
Archimedes Plutonium
by
waiting to insert the concept of infinity borderline, hoping that one
last
gasp of using only the Old-Math fuddy duddy definitions or
nondefinitions of
finite versus infinite. Some of the OldMath conjectures still had
proofs inside
OldMath with resort to infinity borderline. But in the case of
thousands of
conjectures asking whether a set of primes is finite or infinite, all
of them
need that precision definition of infinity.
So let me outline what that proof technique is. We take for granted
the border
of infinity is 10^603, so that if we have that many primes cardinality
we have an
infinite set of primes. And we take for granted that we can search
through to 10^1206
to retrieve primes of that algebraic form or sequence.
So when we say the Regular primes are infinite, what we mean by that
claim is that we
have a container scoop that holds 10^603 primes in that scoop, and we
scoop into all the
numbers of 10^1206 and if that scoop ends up with 10^603 primes or
more from that bin that
holds 10^1206 Natural Numbers, means that form of primes is infinite.
Likewise, if we wanted to know if Mersenne primes or Fermat primes or
factorial-primes, or primes of form
n^2+ 1 was an infinite set of primes, we apply that scoop into all the
Natural Numbers from
0 to 10^1206 and if that scoop comes out full of 10^603 primes of that
specific form, then they are infinite sets of primes of that form. If
the scoop, for example is looking at Mersenne primes and the scoop
ends up at most with 10^200 Mersenne primes within all the numbers of
0 to 10^1206, means of course, that Mersenne primes are a finite set.
Now the reader can understand that given any conjecture of whether a
set is finite or infinite, that it is solvable within the hour by
someone who can gauge the rate of change
of those primes in a sequence layout, for if the distance separating
the neighbors in that sequence is too large of a separation that by
the time we meet up with the number 10^1206 that it is impossible to
have a scoop full of 10^603 of those primes.
The reader can begin to see that this technique works on all sets
whose question is whether the set is infinite or finite.
And the reason OldMath could never conquer any of these conjectures
and allowed the piling up of over thousands of sets begging for a
answer as to whether finite or infinite, is because OldMath never
defined what they meant by finite versus infinite.
In New Math we recognize the deficiencies of Old Math and we inspect
Circumferencing of the Perimeter to see that at 10^-603 with pi having
three zero digits in a row not allowing all finite square perimeters
to be matched by a finite circle circumference, that this number of
10^603 is the border between finite and infinity. And because no
border was given in Old
Math, they started to pile up a back log of thousands of conjectures
that could never be
proven in Old Math.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
> In one sense, I staged the search for Mersenne and n^2+1 primes proof,
> by
> waiting to insert the concept of infinity borderline, hoping that one
> last
> gasp of using only the Old-Math fuddy duddy definitions or
> nondefinitions of
> finite versus infinite. Some of the OldMath conjectures still had
> proofs inside
> OldMath with resort to infinity borderline. But in the case of
correction, that should have been "without resort", and corrected in
original
> Archimedes Plutoniumhttp://www.iw.net/~a_plutonium/
mjc
A minor detail:
It is not necessarily true that both W+1 and W-1 are primes.
All that this shows is that W+1 and W-1 are not divisible by any of
the primes making up W. They may be (and often are) divisible by other
primes.
Too bad.
Archimedes Plutonium
gigantic bin of numbers holding 0 and
all the Naturals to 10^1206. I said that a set of specialized primes
such as Mersenne or Fermat primes
or N^2+1 primes were an infinite set if that scoop shovel could
extract 10^603 of those type of primes
out of the 10^1206 bin.
So this gives us a very easy way of determining whether any set is
finite or infinite. It is easy to see
that even numbers are infinite because we can extract 10^603 even
numbers out of 0 to 10^1206. It is easy
to see that the Regular Primes are an infinite set because there are
10^599 regular primes from 0 to 10^603 and
all we have to do is go to 10^608 to get a full batch of 10^603 primes
and thus infinite.
Now we ask, what is the growth rate of Mersenne primes (2^p)-1?
Mersenne Primes sequence looks like this: 3 ,
7 ,
31 ,
127 ,
8191 ,
131071 ,
54524287, ..
So we ask, how many of Mersenne Primes exist from 0 to 10^1206? Is it
a batch of 10^603 or more? I doubt it.
In fact I doubt it makes a batch of 10^200 Mersenne Primes from 0 to
10^1206 due to the growth rate of Mersenne
primes. In regular primes there are 168 primes from 0 to 1000 yet only
4 Mersenne primes in that same interval, and
the thinning gets much much worse. Based on the rate of thinning we
can estimate whether at 10^1206 we can retrieve
10^603 Mersenne primes out of that interval from 0 to 10^1206. I doubt
we can even get 10^200 Mersenne primes existing in that interval.
And this is how a measurement of whether a set is finite or infinite,
ought to be conducted. A measurement of the density of a object and
whether it thins out the longer we go or whether it can reach a border
of a specified number.
Mathematicians should have recognized a long time ago, that there will
be a mountain of conjectures all wanting to
know if a set of primes is finite or infinite, and they should have,
thence realized, that there must be one technique that will cover that
question with an answer. They should have realized that this mountain
of conjectures is not going to have thousands of disparate techniques
solving a isolated conjecture. Instead, they should have
realized one technique solves all the conjectures concerning the
question of finite or infinite.
Now I am going to hazard a guess as to how many Mersenne primes,
Fermat primes and primes of form N^2+1
exist from 0 to 10^1206.
Mersenne primes roughly 10^200 exist in the interval 0 to 10^1206
Fermat primes roughly 10^180 exist in the interval 0 to 10^1206
Primes of form N^2+1, roughly 10^150 exist in the interval 0 to
10^1206
P.S. I would be interested in knowing at what Polignac even spaced
primes is a finite
set ending the string of infinite sets. This one may surprize me,
because, it maybe a
very high number such as 10^590 spaced primes. Because we can go
through all the numbers
out to 10^1206 to retrieve a scoop of 10^603 such primes.
Transfer Principle
<plutonium.archime...@gmail.com> wrote:
> Now I am going to hazard a guess as to how many Mersenne primes,
> Fermat primes and primes of form N^2+1
> exist from 0 to 10^1206.
> Mersenne primes roughly 10^200 exist in the interval 0 to 10^1206
Actual number of Mersenne primes in that interval: 18. In fact,
here's a list of all 18 of them:
3
7
31
127
8191
131071
524287
2147483647
2^61-1
2^89-1
2^107-1
2^127-1
2^521-1
2^607-1
2^1279-1
2^2203-1
2^2281-1
2^3217-1
Indeed, it's evident that there can't be more than 1206/log(2)
(which is about 4007) Mersenne primes in that interval, since
there are only that many Mersenne _numbers_ in that interval,
much less Mersenne _primes_.
See the following Wikipedia link for more information:
http://en.wikipedia.org/wiki/Mersenne_prime
> Fermat primes roughly 10^180 exist in the interval 0 to 10^1206
Actual number of Fermat primes in that interval: 5. In fact, here
is a list of all five of them:
3
5
17
257
65537
4294967297
It's evident that there can't be more than log(4007)/log(2)
(which is about 12) Fermat primes in that interval, since
there are only that many Fermat _numbers_ in that interval,
much less Fermat _primes_.
See the following Wikipedia link for more information:
Archimedes Plutonium
Yes, I was not thinking when I made those wild guesses. Thinking more
of the Polignac
even numbered primes.
> Indeed, it's evident that there can't be more than 1206/log(2)
> (which is about 4007) Mersenne primes in that interval, since
> there are only that many Mersenne _numbers_ in that interval,
> much less Mersenne _primes_.
>
The game I play is that I look to see how many Mersenne are in the 0
to 100
interval and then create an upper bound formula. Since there are only
three Mersennes
then I have Ln(100)= 4. So then I take Ln(10^1206) for a maximum
Mersennes from 0
to 10^1206 is roughly 2450
> See the following Wikipedia link for more information:
>
> http://en.wikipedia.org/wiki/Mersenne_prime
>
> > Fermat primes roughly 10^180 exist in the interval 0 to 10^1206
>
Another laughable wild guess or gaffe.
> Actual number of Fermat primes in that interval: 5. In fact, here
> is a list of all five of them:
>
> 3
> 5
> 17
> 257
> 65537
> 4294967297
>
> It's evident that there can't be more than log(4007)/log(2)
> (which is about 12) Fermat primes in that interval, since
> there are only that many Fermat _numbers_ in that interval,
> much less Fermat _primes_.
>
> See the following Wikipedia link for more information:
>
> http://en.wikipedia.org/wiki/Fermat_prime
Alright,the game I play is that there are 3 in the 0 to 100 interval
gives
me an upper bound formula of again Ln(100) = 4. So that Ln(10^1206) is
roughly
2450.
Both of them way too small to be a cardinality set of 10^603 and thus
those are finite sets.
Have you found anything on N^2 +1 primes?
My game would be to see how many in the 0 to 100 interval and bound it
on top by a formula.
I see that these primes are 2, 5, 17, 37, . . which is Ln(100) = 4
which suggests that
0 to 10^1206 has Ln(10^1206) or approx 2450 such primes and again a
finite set of primes.
Now if any mathematicians bulks at the notion that a finite set is one
that cannot retrieve a set of 10^603 cardinality of those primes out
of a field of 10^1206, well,they can easily slide the measuring scale
further down, by saying 10^1000 is the borderline of infinity and
allowed to use 10^2000 for algebra completeness and will thus end up
with the same conclusion that twin primes are infinite, primes are
infinite, but Mersenne primes, Fermat
primes, and n^2+1 primes are finite based on this infinity borderline
concept.
LWalk, I am trying to think of that algebraic form of primes that
looks as though they are an infinite set of primes, but in actuality
they are finite. Is it primes of form 3k +1 ?
Or is it primes of form 2^k +1? Do you remember any forms that looked
infinite but were finite?
Another question LWalk. Granted that infinity border is 10^603 and
allowed to go up to 10^1206. Consider the Polignac Conjecture which
has twin primes, quad, hex etc etc.
Now the twin primes have I am guessing, half of 24 or 12 in the primes
from 0 to 100.
So that would be a formula of x/ 2(Ln)x = 12 for an upper bound. This
would mean that
there are roughly 10^1200 primes in 10^1206 and roughly 10^1198 twin
primes in the interval
0 to 10^1206. The question is, at what Polignac even spaced primes are
there only 10^600
of those primes in the interval 0 to 10^1206? Is it something like two
primes P and Q where
P-Q = 10^300, any insights??
Archimedes Plutonium
hex separated by 6,
etc primes.
When Infinity borderline is 10^603 and we can go out to 10^1206 for
the completeness of
algebra, then we must examine the question of when the Polignac primes
become finite and
when they stay infinite.
These are the first 25 regular primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97
There are 15 twin primes in that interval:
3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, 59, 61, 71, 73
There are 15 quad primes in that interval:
3, 7, 11, 13, 17, 19, 23, 37, 41, 43, 47, 67, 71, 79, 83
There are 21 hex primes in that interval:
5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 73,
79, 83, 89
There are 18 octo primes in that interval:
3, 5, 11, 13, 17, 19, 23, 29, 31, 37, 53, 59, 61, 67, 71, 79, 89, 97
There are 19 deca primes in that interval:
3, 7, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 71, 73, 79, 83,
89
The Polignac primes turn out to be counterintuitive and very
surprizing. Our imagination
paints a different picture than actual reality. The primes sets
increase in membership, whereas I thought they decrease.
There are 12 [50s] primes in that interval:
3, 11, 17, 23, 29, 47, 53, 61, 67, 73, 79, 97
So let me mull over these results. I know that some of these Polignac
primes are finite while the remainder are infinite given the precision
definition of Infinity as 10^603.
And this is a result totally unexpected by the rest of the mathematics
community who was
entrenched in the thought that either the entire lot is finite or the
entire lot is infinite.
Archimedes Plutonium
And, perhaps the entire math community, entrenched as it is, was
correct that
the entire lot is infinite, considering that we can play with all the
numbers
up to 10^1206 when the border is 10^603.
So in the Polignac conjecture if I suppose the border is 100 and can
go out as far as
10000, means that I can furnish [100s] primes.
So it looks as though the entrenched math community wins one over AP
on this scoreboard
of Polignac primes.
But it still all seems counterintuitive.
Archimedes Plutonium
Is there a prime connection to pi, then, due to Polignac primes, that
they are all
infinite sets of primes? Is there some connection with the fact that
pi has three zero
digits in a row at 10^-603 wherein for the first time a finite
perimeter is not matched
by a finite circumference at 10^603? Does the three zero digits in a
row have a connection
with how many Polignac primes are contained in 10^603 and its
algebraic domain of 10^1206?
What I am trying to find out is what the Prime relationship is to the
fact that pi has
three zero digits in a row at 10^603. That relationship maybe the
orderliness of the Polignac primes, that they are the utmost order if
the border was taken to be 10^603 and not something else.
So I want to keep my eyes always open for a relationship of the primes
that I can say "yes indeed, the primes have to be like that because pi
has three zero digits in a row at 10^603".
Archimedes Plutonium
So far, that relationship of pi to primes is only seen in the Wallis
series:
pi = 4- 4/3 + 4/5 - 4/7 + ...
And that is just not enough of a view into pi and primes. We need to
know
how the primes can predict that the number 10^603 or 10^-603 is a
borderline
of infinity. We see how pi alone predicts the borderline by yielding a
finite
perimeter of a square not able to be matched by a finite circle
circumference.
So how does the primes alone show us the border is 10^603? Perhaps
these Polignac primes
can open a door or window.
Archimedes Plutonium
supposing
100 was the borderline of infinity which allows me to go out to 10,000
for Algebra completeness. I want to see if there are 100 primes of the
[100s polignac primes] from 0 to 10,000. If there are, then that would
be supporting evidence that there are at least 10^603 primes of
[10^603
polignac primes] from 0 to 10^1206.
If there are, then the Polignac Conjecture is likely to be true as
infinite
sets of all the Polignac primes.
--- quoting from http://primes.utm.edu/lists/small/1000.txt ---
2 3 5 7 11 13 17 19 23
29
31 37 41 43 47 53 59 61 67
71
73 79 83 89 97 101 103 107 109
113
127 131 137 139 149 151 157 163 167
173
179 181 191 193 197 199 211 223 227
229
233 239 241 251 257 263 269 271 277
281
283 293 307 311 313 317 331 337 347
349
353 359 367 373 379 383 389 397 401
409
419 421 431 433 439 443 449 457 461
463
467 479 487 491 499 503 509 521 523
541
547 557 563 569 571 577 587 593 599
601
607 613 617 619 631 641 643 647 653
659
661 673 677 683 691 701 709 719 727
733
739 743 751 757 761 769 773 787 797
809
811 821 823 827 829 839 853 857 859
863
877 881 883 887 907 911 919 929 937
941
947 953 967 971 977 983 991 997 1009
1013
1019 1021 1031 1033 1039 1049 1051 1061 1063
1069
1087 1091 1093 1097 1103 1109 1117 1123 1129
1151
1153 1163 1171 1181 1187 1193 1201 1213 1217
1223
1229 1231 1237 1249 1259 1277 1279 1283 1289
1291
1297 1301 1303 1307 1319 1321 1327 1361 1367
1373
1381 1399 1409 1423 1427 1429 1433 1439 1447
1451
1453 1459 1471 1481 1483 1487 1489 1493 1499
1511
1523 1531 1543 1549 1553 1559 1567 1571 1579
1583
1597 1601 1607 1609 1613 1619 1621 1627 1637
1657
1663 1667 1669 1693 1697 1699 1709 1721 1723
1733
1741 1747 1753 1759 1777 1783 1787 1789 1801
1811
1823 1831 1847 1861 1867 1871 1873 1877 1879
1889
1901 1907 1913 1931 1933 1949 1951 1973 1979
1987
1993 1997 1999 2003 2011 2017 2027 2029 2039
2053
2063 2069 2081 2083 2087 2089 2099 2111 2113
2129
2131 2137 2141 2143 2153 2161 2179 2203 2207
2213
2221 2237 2239 2243 2251 2267 2269 2273 2281
2287
2293 2297 2309 2311 2333 2339 2341 2347 2351
2357
2371 2377 2381 2383 2389 2393 2399 2411 2417
2423
2437 2441 2447 2459 2467 2473 2477 2503 2521
2531
2539 2543 2549 2551 2557 2579 2591 2593 2609
2617
2621 2633 2647 2657 2659 2663 2671 2677 2683
2687
2689 2693 2699 2707 2711 2713 2719 2729 2731
2741
2749 2753 2767 2777 2789 2791 2797 2801 2803
2819
2833 2837 2843 2851 2857 2861 2879 2887 2897
2903
2909 2917 2927 2939 2953 2957 2963 2969 2971
2999
3001 3011 3019 3023 3037 3041 3049 3061 3067
3079
3083 3089 3109 3119 3121 3137 3163 3167 3169
3181
3187 3191 3203 3209 3217 3221 3229 3251 3253
3257
3259 3271 3299 3301 3307 3313 3319 3323 3329
3331
3343 3347 3359 3361 3371 3373 3389 3391 3407
3413
3433 3449 3457 3461 3463 3467 3469 3491 3499
3511
3517 3527 3529 3533 3539 3541 3547 3557 3559
3571
3581 3583 3593 3607 3613 3617 3623 3631 3637
3643
3659 3671 3673 3677 3691 3697 3701 3709 3719
3727
3733 3739 3761 3767 3769 3779 3793 3797 3803
3821
3823 3833 3847 3851 3853 3863 3877 3881 3889
3907
3911 3917 3919 3923 3929 3931 3943 3947 3967
3989
4001 4003 4007 4013 4019 4021 4027 4049 4051
4057
4073 4079 4091 4093 4099 4111 4127 4129 4133
4139
4153 4157 4159 4177 4201 4211 4217 4219 4229
4231
4241 4243 4253 4259 4261 4271 4273 4283 4289
4297
4327 4337 4339 4349 4357 4363 4373 4391 4397
4409
4421 4423 4441 4447 4451 4457 4463 4481 4483
4493
4507 4513 4517 4519 4523 4547 4549 4561 4567
4583
4591 4597 4603 4621 4637 4639 4643 4649 4651
4657
4663 4673 4679 4691 4703 4721 4723 4729 4733
4751
4759 4783 4787 4789 4793 4799 4801 4813 4817
4831
4861 4871 4877 4889 4903 4909 4919 4931 4933
4937
4943 4951 4957 4967 4969 4973 4987 4993 4999
5003
5009 5011 5021 5023 5039 5051 5059 5077 5081
5087
5099 5101 5107 5113 5119 5147 5153 5167 5171
5179
5189 5197 5209 5227 5231 5233 5237 5261 5273
5279
5281 5297 5303 5309 5323 5333 5347 5351 5381
5387
5393 5399 5407 5413 5417 5419 5431 5437 5441
5443
5449 5471 5477 5479 5483 5501 5503 5507 5519
5521
5527 5531 5557 5563 5569 5573 5581 5591 5623
5639
5641 5647 5651 5653 5657 5659 5669 5683 5689
5693
5701 5711 5717 5737 5741 5743 5749 5779 5783
5791
5801 5807 5813 5821 5827 5839 5843 5849 5851
5857
5861 5867 5869 5879 5881 5897 5903 5923 5927
5939
5953 5981 5987 6007 6011 6029 6037 6043 6047
6053
6067 6073 6079 6089 6091 6101 6113 6121 6131
6133
6143 6151 6163 6173 6197 6199 6203 6211 6217
6221
6229 6247 6257 6263 6269 6271 6277 6287 6299
6301
6311 6317 6323 6329 6337 6343 6353 6359 6361
6367
6373 6379 6389 6397 6421 6427 6449 6451 6469
6473
6481 6491 6521 6529 6547 6551 6553 6563 6569
6571
6577 6581 6599 6607 6619 6637 6653 6659 6661
6673
6679 6689 6691 6701 6703 6709 6719 6733 6737
6761
6763 6779 6781 6791 6793 6803 6823 6827 6829
6833
6841 6857 6863 6869 6871 6883 6899 6907 6911
6917
6947 6949 6959 6961 6967 6971 6977 6983 6991
6997
7001 7013 7019 7027 7039 7043 7057 7069 7079
7103
7109 7121 7127 7129 7151 7159 7177 7187 7193 7207
7211 7213 7219 7229 7237 7243 7247 7253 7283
7297
7307 7309 7321 7331 7333 7349 7351 7369 7393
7411
7417 7433 7451 7457 7459 7477 7481 7487 7489
7499
7507 7517 7523 7529 7537 7541 7547 7549 7559
7561
7573 7577 7583 7589 7591 7603 7607 7621 7639
7643
7649 7669 7673 7681 7687 7691 7699 7703 7717
7723
7727 7741 7753 7757 7759 7789 7793 7817 7823
7829
7841 7853 7867 7873 7877 7879 7883 7901 7907
7919
--- end quoting from http://primes.utm.edu/lists/small/1000.txt ---
Actually the above does not go to 10,000 but I am hoping that there
are 100
of those [100s polignac primes] in the above 7919 and for me not to
have
to look for the rest of the primes up to 10,000.
The first 100 of the [100s polignac primes]
3, 103, 7, 107, 13, 113, 31, 131, 37, 137,
67, 167, 97 , 197, 127, 227, 139, 239, 151, 251,
157, 257, 163, 263, 181, 281, 193, 293, 211, 311,
283, 383, 331, 431, 349, 449, 367, 467, 379, 479,
409, 509, 421, 521, 457, 557, 463, 563, 487, 587,
499, 599, 541, 641, 547, 647, 577, 677, 601, 701,
619, 719, 643, 743, 661, 761, 673, 773, 709, 809,
727, 827, 739, 839, 757, 857, 787, 887, 811, 911,
829, 929, 853, 953, 877, 977, 883, 983, 919, 1019
991, 1091, 997, 1097, 1009, 1109, 1051, 1151, 1063, 1163
So I did not need to go very far away from 100 to pick up
100 of the [100s polignac primes]
Much the same story can be said for the [10^603s polignac primes]
to be hunted down in the territory of 10^603 to that of 10^1206
So that is an informal proof that the Polignac primes are all infinite
sets.
And it also bolsters the case of where I convert the Twin Primes proof
into
proving Quad and Hex primes infinity and where that same proof
schemata proves
all Polignac primes as infinite sets.
So here I have two proof methods of proving the Polignac conjecture,
one of the Twin Primes
format and one of the above listing showing that [10^603s polignac
primes] are infinite.
P.S. my intuition was fouled up in starting this Polignac but once I
do the actual listing of
primes, the picture becomes very clear.
And it also shows how infinity is thought of more as density, such as
in the case of Mersenne primes
that there are only a handful from 0 to 10^1206, but that the Polignac
primes are almost as frequent
as are the regular primes.
P.P.S. Now I did not escape noticing of the plethora of HEX primes in
the Polignac conjecture. Further questions would be whether those hex
primes are the most abundant primes of all the Polignac primes and why
would that be the case? Also, perhaps only at the juncture of 10^603,
is it possible that all the Polignac primes sort of come together in
some sort of equilibrium of number cardinality? That equilibrium maybe
the mirror reflection of the fact that pi at 10^-603 has three zero
digits in a row? So I am always looking for the connection of pi at
10^603 and the primes at 10^603.
Archimedes Plutonium
My focus now is on density of primes and to see what is the smallest
infinite set of primes
given that the Regular primes are what they are. I need to sequence
the primes of form
n^2+1 and of n^2-1 to see if they can have extra terms in order to
make them the smallest
infinite set. Of course, when infinity border is 10^603 allowed
10^1206 for algebraic completeness
that the Mersenne primes and Fermat primes and primes of form n^2+1
are proven to be finite sets.
Which begs the question that there is a algebraic form that is what
can be described as a smallest
infinity set of primes. Examining the Polignac primes of 100 border
and algebraic completeness at
10^4, that I would have to go to about 8,000 evenly spaced primes to
fetch me a lot of 100 such primes
and thus infinite. So it maybe the case that the smallest infinite set
of primes follows a formula of this:
x/Ln(x) divided by Ln(x)
So that in the case of 10,000 there are ten groups of primes each
containing 100 cardinality of primes.
Maybe this formula or function is going to be the best function for
the smallest infinity sequence or set:
(x/Ln(x)) / Ln(x)
But that formula is not as nice or neat as n^2+1 so I want to try to
find a nice neat sequence without the logarithm
involved.
spudnik
in an AMS publ., very short but, of course,
it does not generate all of the prime,
like what's-his-name's 26-degree equation.
Archimedes Plutonium
3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433,
449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757,
35999, 37511, 50833, 81839.
And if they only went 4X more in the interval 0 to 10,000 that they
would be a infinite set. I counted 26 in the
interval. Which looks as though the Fibonacci primes are a microcosm
of the Regular primes following the law
of x/Ln(x) within that specific Fibonacci sequence.
But according to the rules of mathematics, when infinity border is
10^603 with its algebraic completeness
at 10^1206, that the Fibonacci primes and the Mersenne primes and the
Fermat primes and primes of form n^2 +1
are all finite sets, for their density winnows out rapidly so that
there cannot be a set of 10^603 cardinality contained inside of
10^1206.
So I am looking for the smallest form of primes for a smallest
infinity set. It must be somewhere between
the Fibonacci primes because they were off, only by a factor of 4, and
between Twin primes.
Archimedes Plutonium
course mathematical
induction can be used since we cannot list all the primes to 10^1206
as algebraic completeness of the numbers from 0 to 10^603.
Apparently primes of this form N^2 +1 have not been given a fancy name
like Mersenne primes
or Fermat's primes or Sophie Germaine primes which are also finite
sets.
2, 5, 17, 37, 101, 197, 257, 401, 577, 677, 1297, 1601, 2917, 3137,
4357, 5477, 7057, 8101, 8837, 12101, 13457, 14401, 15377, 15877,
16901, 17957, 21317, 22501, 24337, 25601, 28901, 30977, 32401, 33857,
41617, 42437, 44101, 50177
Suppose 100 was the borderline of infinity and its algebraic
completeness at 10^4. We simply see if there are 100 primes of form
N^2 +1 in the interval 0 to 10^4, and if so
then the set is infinite, and if not then the set is finite, and ditto
for 10^603 with
10^1206.
From my count above there are only 19 such primes in that interval 0
to 10^4, which means those primes are finite.
So the above is a technique that conquers and vanquishes all those
Number theory conjectures worried about whether a specific set of
primes is finite or infinite. I suppose the two most important of
those conjectures are the ones that relate to geometry such as the
Sophie Germain primes and
Fibonacci primes, and the Fermat primes and the Mersenne primes
(related to perfect numbers). All four of those primes are proved
Finite with the above technique.
Also,the primorial primes and the factorial primes are finite sets due
to their rapid rise
and low density.
What Old Math failed to do is realize that Infinity and Finiteness has
a border to cross,
and once you find that 10^603 is that border due to pi having three
zero digits in a row
and causing some finite perimeters unable to be matched with a finite
circle circumference
and using 10^1206 we easily prove all these long past conjectures, and
we free up all those
mathematicians who bark out more of these conjectures questioning
finite versus infinite of
prime sets. Because as soon as they bark out their conjecture, the
technique is able to answer it.
Now I still am wondering if there is a smallest function or algorithm
for the smallest infinite set of primes and the largest finite set of
primes?
Now I know that Euler gave a function a long time ago of n^2 + n +41
that accurately gives the primes for n<40. I also know that there is
no formula to encapsulate all the primes, similar to the idea that pi
is not algebraic.
But I wonder if there is a new formula, a modern day Euler for that
using the idea that
infinity is the borderline at 10^603 or 10^1206 so that we can have a
function such as
n^2 + n + 10^603 with a specified n.
What I am trying to fathom is a smallest Infinite set of primes, for
we see that Twin Primes is infinite and smaller than Regular primes
and we see that N^2+1 primes is finite
and smaller than Fibonacci Primes. So we have a ranking of size of
Finite sets and Infinite
sets of primes so one can expect to find some function which although
does not capture all primes but a sequence of primes which is the
smallest function of infinity set of primes.
Now I do not mean that we can simply prefabricate a set that contains
10^603 primes exactly
and having gone all the way out to 10^1206 to retrieve them, because
we find there is no
function to represent that specific sequence that we hand picked.
So we see, here in mathematics that we have sequences for which we can
not have a function that represents that sequence, nor an algorithm
that represents that sequence.
I am not sure what Old-Math thought of this situation? Whether they
thought that every sequence is somehow representable as a algorithm or
a function, is up in the air. They did
insist that there is no function to represent all the primes, but
since some functions represent primes perfectly well such as Mersenne
primes, one has to wonder where representation ends and where it
exists. Maybe the gulf or divide between primes of form Fibonacci
primes as finite and twin primes is a gulf that is not able to be
improved upon in seeking a smallest infinite set of primes, even
though we can go through the entire prime list to 10^1206 and hand
pick out the last 10^603 primes. Maybe we can do that theoretically,
but never practically, and never have a formula, function or algorithm
for those hand picked 10^603 primes ending at 10^1206.
Archimedes Plutonium
questioning whether
a set is finite or infinite. The imbalance is that these questions for
the most
part involve some subclass of primes. We hardly ever hear of a
conjecture over the
infinitude of some geometry object or some other numbers, other than
primes. So in
mathematics, if there is a conjecture about whether finite or
infinite, it usually
is a question over some subclass of primes.
I want to get away from primes in this post and focus this new-found
technique
of proving whether finite or infinite on something other than primes
subclasses.
So tonight I want to pick on perfect-squares and on Champernowne type
of numbers.
Now Old Math would never even put these to some test of finite or
infinite because
in Old Math, finite and infinite were never really precisely defined.
When you have
no definition, you do not go chasing after something to see if it
obeys anything.
If we do not have any understanding or definition of magnetism, we do
not go and look
or measure for magnetism. Same thing happened in Old Math, that no-one
was going to ask
are the set of Perfect Square Numbers finite or infinite, well, of
course they would never ask that because they never had a precision
definition of infinite and they automatically assumed Perfect Square
Numbers are infinite.
But the reason I pick on Perfect Square Numbers is because they form a
smallest infinity set in the Natural Numbers. This set sequence is 1,
4, 9, 25, 36, 49, . .
So now, let me apply the technique of proving finite or infinite set.
I will prove in via
mathematical induction with the borderline of infinity at 10^603 and
its algebraic completeness at 10^1206. To prove it infinite I need to
show for the case of 100 and its
algebraic completeness of 10^4 that there are 100 perfect square
numbers in 0 to 10^4.
And obviously there are since 10^4 is 100^2.
Now I bring up this discussion of Perfect Squares because I am looking
for the smallest infinity set of primes. For all the Natural Numbers,
there is a smallest infinity set as
represented by the Perfect Square set. Its density is just right, not
too many and not one
few to be an infinite set. And for 10^603 to its algebraic
completeness of 10^1206 there are exactly 10^603 Perfect Squares in
that set, and thus infinite.
But now let me show you a set that Old Math presumed was infinite,
only because they never
precisely defined infinity with a border crossing from finite.
--- quoting Wikipedia on Champernowne type numbers ---
In mathematics, the Champernowne constant C10 is a transcendental real
constant whose decimal expansion has important properties. It is named
after mathematician D. G. Champernowne, who published it as an
undergraduate in 1933.
In base 10, the number is defined by concatenating successive
integers:
C10 = 0.12345678910111213141516...
--- end quoting ---
Now we consider Champernowne type numbers as the following sequence:
1, 12, 123, 1234, 12345, 123456, etc etc
Now in Old Math that sequence composed an infinite set of numbers. In
NewMath
with infinity and finite precisely defined the above sequence of
numbers and that set is
finite. It is finite because from 100 to 10^4 there are only 4 members
when it requires
100 members. And from 0 to 10^1206 there are no 10^603 of those
numbers contained, hence the set is finite.
So in NewMath, we have to go back and review a lot of subjects and
material which we previously thought were infinite sets, when in fact
they were smallishly finite sets.
Even the concept of Champernowne's constant as some form of
transcendental nature was poppycock fantasy when math has a precision
definition of infinity.
Archimedes Plutonium
There is a very intriguing result to Algebra when finite versus
infinity is given a precision definition.
When mathematics has a border crossing from finite to infinite it does
a very nice and strange affect
on Algebra. And shows why Old-Math algebra was seriously flawed.
In my last posts I accounted for why Perfect Square Numbers were an
infinite set:
1, 4, 9, 16, 25, 36, . .
They are an infinite set because the border of finite with infinity is
10^603 and its Algebraic
Completeness is 10^1206, where we have exactly 10^603 cardinality of
Perfect Squares residing
in that Field of 0 to 10^1206.
However, Perfect Cubes of this sequence:
1, 8, 27, 64, . .
Is a Finite set, because one cannot extract 10^603 cardinality of
perfect cubes out of 0 to 10^1206.
Now most will immediately complain about I have done above, saying it
is ad hoc. But is it really ad hoc
to say that to complete Algebra of its add, subtract, multiply and
divide, is a duality operation. If the border
of infinity is 10^603, then the most that multiplication can do is
10^603 x 10^603 = 10^1206 and the most
that division could ever be is 10^-603 / 10^603 = 10^-1206. And is it
not true that cubing is nothing more than
a reiteration of squaring, where 3x3 = 9 and where (3x3) x3 = 27?
So that Algebraic Completeness does not mean completeness of exponent,
but only completeness of the operators
add, subtract, multiply, divide in a bi-number operation and anything
further is a reiteration of a bi-number
operation.
So when I discovered that 10^603 is the border of finite into infinity
because pi has three zero digits in a row
at 10^-603 which disallows the existence of a finite circumference to
match a finite perimeter, then the algebraic
completeness for add, subtract multiply and divide needed to go only
to 10^1206 to have a completeness of
those operators.
In Old-Math, they never realized that finite must have a border with
infinity. They never looked for a border crossing.
They never precision defined the term infinite in Old-Math.
Consequently, their algebra also suffered in that they assumed
Algebraic Completeness was as open-ended as was not having a precision
definition of infinity.
In Old-Math, they believed the set of Perfect Squares had the
identical cardinality as the set of Perfect Cubes and the
set of Perfect 10^1000 objects. A major flaw of Old Math Algebra, is
that such a system ill-defines the concept of
a "mathematical operator" such as addition or multiplication or
subtraction or division. Those operations are bi-number
operators and cubing is just a reiteration of squaring. In Old-Math
Algebra, we have the illogic of an infinitude of operators in
mathematics. In New Math, since finite and infinity are well-defined,
the operators of Algebra are finite.
Archimedes Plutonium
finite versus infinite.
We need that border in order to precisely define infinity from
finiteness and you cannot
have the two related to each other without that border that is crossed
from finiteness and
entering infinity. So the borderline is the precision definition of
finiteness and infinity.
But the mystery arises in that as you discover this borderline, that a
second border automatically springs up that is the border of algebraic
completeness. So we discover
the natural border of finite with infinity as 10^603 where pi has
three zero digits in
a row and where a finite perimeter is unmatched by a finite circle
circumference.
So up springs a natural borderline of finite versus infinite at
10^603, but instantly another borderline springs up as the Algebraic
Completeness border.
So the numbers of mathematics really stops being trustworthy at 10^603
and we can say that
mathematics ends at 10^603 as being reliable Aristotelian Logic and
beyond 10^603 is nebuluous math. But we can use all the numbers from 0
to 10^1206 to ensure the Algebraic
side of mathematics. So this is a mystery that the birth of the
borderline between finite
and infinity, gives automatic birth to the square of that borderline
as the Algebraic Completeness.
In physics we call this relationship a quantum duality, that you need
both and that one alone cannot exist without the other, and in physics
this is particle wave nature of reality.
Archimedes Plutonium
book, but as for now I stick it into the
infinitude of primes chapter.
As I have be saying for the past two decades, that unless someone can
actually translate this conjecture from physics and electrical
engineering into a well defined mathematical problem, it is pseudo
math until that time.
This is the first conjecture posed as mathematical when in fact it is
physics and engineering. It involves "time"
which no other mathematical statement involved itself with the "time
of physics."
--- quoting Wikipedia on what the P versus NP problem is ---
Consider the subset sum problem, an example of a problem that is easy
to verify, but whose answer may be difficult to compute. Given a set
of integers, does some nonempty subset of them sum to 0? For instance,
does a subset of the set {-2, -3, 15, 14, 7, -10} add up to 0? The
answer "yes, because {-2, -3, -10, 15} add up to zero" can be quickly
verified with three additions. However, finding such a subset in the
first place could take more time; hence this problem is in NP (quickly
checkable) but not necessarily in P (quickly solvable).
An answer to the P = NP question would determine whether problems like
the subset-sum problem that can be verified in polynomial time can
also be solved in polynomial time. If it turned out that P does not
equal NP, it would mean that some NP problems are harder to compute
than to verify: they could not be solved in polynomial time, but the
answer could be verified in polynomial time.
--- end quoting Wikipedia ---
HISTORICAL ASIDE: I like to ask a history of math question here. I
notice that Wikipedia talks about the P versus
NP problem in two manners, one as P versus NP and another as P=NP. The
question I have is whether my talk through the years of Finite versus
Infinite to find that border, whether the mathematics community
borrowed my habits of calling it "Finite versus Infinite" and borrowed
that "versus" to use for their P versus NP? What I mean is not an
isolated one off case of borrowing but a standard way of using
"versus" in a mathematics context.
So I want to know if I had started that terminology and now others are
using the "versus" for other areas of mathematics like P versus NP
which Wikipedia is using? It could be the case the P versus NP was
much older than when I hit the scenes with Finite versus Infinite.
Now let me zoom far ahead with a prognosis of the P versus NP problem.
Let me say that the P versus NP is like that
of listing all the primes from 0 to 10^603 and to 10^1206. The P
versus NP was never given with a precision definition of infinity
until now. So how does the border of infinity at 10^603 affect the P
versus NP problem. Would it not automatically say that the conjecture
ill-composed as well as being ill-defined? Human time and computer
time could not possibly list all the primes in 10^603 because there
are about 10^600 primes in there, let alone list all the primes
in 10^1206. So the P versus NP problem was never really a mathematical
problem to start with. So there is no way of
well defining concepts as "quick time" nor "computable time" nor
"verifiable time".
Notice however we can compute and verify all the Mersenne primes in 0
to 10^603 and 0 to 10^1206. So we can have
isolated cases of a P=NP, but we cannot have a well-defined conjecture
of a generalized P versus NP.
Again, this is a conjecture where math has overstepped its territory
of what it can and cannot do. It is not a piece of mathematics but a
brain teazer for those in physics and engineering.
Archimedes Plutonium
defined as a cousin to equal or not equal but having a lower status
than equal. It is a term to be used when two items have a relationship
but that relationship is
not yet clearly defined.
So I am wondering if Archimedes Plutonium is the first in mathematics
history to use the term "versus" in a
mathematical argument with that above definition. I am wondering that
the Finite versus Infinite defining process
needs a math term of "versus" since there is no other math term
appropriate for that description. We cannot say
Finite equal or not equal to Infinity, but need a term to describe
them as related such as "versus".
Now I see that the P and NP problem is using the term "versus". It
maybe the case that I subconsciously borrowed from an already existing
term of " versus" in the math literature. Or it maybe the case that I
was the first to use
the term "versus" and made it a mathematical term. I can easily make a
google search in my old posts to when I began
heavily using the term "versus".
Here is Wikipedia using the term "versus"
--- quoting Wikipedia ---
The P versus NP problem is a major unsolved problem in computer
science. Informally, it asks whether every problem whose solution can
be efficiently checked by a computer can also be efficiently solved by
a computer. It was introduced in 1971 by Stephen Cook in his paper
"The complexity of theorem proving procedures"[2] and is considered by
many to be the most important problem in the field.[3] It is one of
the seven Millennium Prize Problems selected by the Clay Mathematics
Institute to carry a US$ 1,000,000 prize for the first correct
solution.
In essence, the question P = NP? asks:
Suppose that solutions to a problem can be verified quickly. Then, can
the solutions themselves also be computed quickly?
The theoretical notion of quick used here is an algorithm that runs in
polynomial time. The general class of questions for which some
algorithm can provide an answer in polynomial time is called "class P"
or just "P".
--- end quoting ---
When I was looking for the borderline between Finite and Infinity I
could not say equal or greater than or less than or any other
mathematical term, so I struck on the idea that the term "versus" is
neutral enough yet still conveys the
idea that finite is related to infinity.
Same thing goes for P and NP problem. Now the natural choice of a term
may have been "and" and somewhere the
P NP picked up the term versus.
So I wonder if I have priority discovery rights of using the term
"versus" in mathematics history for a term that means
two things are related, but we just do not know the full extent of the
relationship, or that we want to have a term that
simply says two things are related.
So does anyone know if AP is the first mathematician to use the term
"versus" as a mathematical term? Or did AP
subconsciously borrow "versus" from older math literature?
Archimedes Plutonium
sort of concept of a relationship, but
a relationship that is not specific but "in general". Such as in the
use of Finite versus Infinite or
in the case of P versus NP. Now other terms could be substituted such
as "and" or "related to" so that
we have P related to NP or Finite related to Infinite. But the term
"versus" has a sort of set characteristic
that Set theory does not yet defines.
Which is better of these three? P related to NP, or P and NP, or P
versus NP? Likewise which conveys the
best meaning of these three? Finite related to Infinite,or, Finite and
Infinite, or, Finite versus Infinite?
In both cases the term "versus" is better suited. For the term
"versus" has the quality of two items related,
yet those two items have a border or separation. One could think of
two disjoint sets that intersect at a borderline.
--- quoting old 1995 post of mine where I used Finite versus Infinite
---
Newsgroups: sci.math, alt.sci.physics.plutonium
From: Archimedes.Pluton...@dartmouth.edu (Archimedes Plutonium)
Date: 1995/10/18
Subject: Re: Finite Integers as goofy as Finite Reals
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In article <4630cp$...@chebyshev.cs.umd.edu>
arte...@cs.umd.edu (Santiago Arteaga) writes:
> >Santi are these two Infinite Integers?
> >. . .9977392256259918212890625
> > . . .0022607743740081787109376
> Yes. So they do not provide a counterexample for Fermat's
> last theorem, which was meant to hold on the set of finite numbers.
> Santi
No Santi, you are saying that since pi can be written by a symbol
like pi that it is finite. That .3333... can be written like 1/3 that
it is finite.
So if I write . . .9977392256259918212890625 as X, then it is finite.
That is what your argument Santi amounts to. Because you, Santi
replace
actual numbers with symbols you want to bypass the whole argument of
finite versus infinite.
What I am saying is that Math is the science of precision. It is
not
precise for you Santi to claim 1 and 7 are Finite Integers and that
1/7
is a Finite Real yet have .14285.... a Finite Real when it has an
infinite string rightwards. You want to get rid of your imprecision
Santi by just calling this number a symbol like A or Y or whatever.
Santi. Do you still say the Finite Natural of 1 is the same as the
Infinite Adic of ...00001? Do you agree with that?
At least we can agree Santi that math is the science of precision.
Or
do you disagree with that statement and that math does not strive for
precision?
--- end quoting older post of mine going to 1995 where I used the math
term "versus" ---
So did AP borrow the term versus already existing in mathematical
terminology, or did I
invent "versus" as a math term starting in the 1990s? Was the P;NP
problem called the P versus
NP problem before the 1990s? Or has the mathematics community borrowed
from AP the term
"versus" after following his Finite versus Infinite debates?
Veky
1. "Time" in context of P and NP is a well defined mathematical concept. It counts steps that Turing machine makes in solving a problem. BTW, "machine" is also a defined concept: it is a discrete system of iterating a certain function. It might be confusing to people without CS background, but it is mostly standard process in math.
2. "versus" is an old Latin word. And no, you in the "finite vs. infinite" do _not_ use it in the same sense as in "P vs. NP", because the relation there is pretty clear: finite is obviously _less than_ infinite.
3. 10^1206+1 is a natural number.
Archimedes Plutonium
ado" over nothing.
But the term "versus" looks like it needs inclusion in mathematics
proper where two items
are sort of rivals and are related.
But what really brings me to this topic of "versus" is that I do not
recall "versus" used
for the P and NP problem before and that I am not really seeking a
priority discovery
right, but rather, I am fascinated of my history of using the terms
Finite versus Infinity
and here I find that others use the terms P versus NP. What I am
trying to say quickly,
is that the P versus NP problem is a disguised problem which is
underlain by the true problem
of Finite versus Infinite. So that the P versus NP problem when put
into its default mode, or
reduced to most simple form such as reducing a fraction, that the P
versus NP was just the Finite versus
Infinite problem. That when you have to place the borderline for P and
NP, and the borderline is
10^603 with a algebraic completeness at 10^1206 that there never was a
P versus NP problem, just a
misunderstanding that you cannot have "efficiently checked" and
"efficiently solved" equal to one another,
because one lays in the 10^603 territory and the other in the 10^1206
territory.
So I am not wasting posting time on priority discovery rights on the
term "versus", but rather
I am curious of this serendipity use of the term "versus" when Finite
versus Infinity is the default
zone of P versus NP. Finite is not equal to Infinite, and so, in the
default of P versus NP, the P is
not equal to NP. So that it should be P=/NP as the solution.
An example is that of Perfect Squares. The Perfect Squares forms an
infinite set because there are
10^603 perfect squares between 0 and 10^1206. So we can instantly
verify or check efficiently that such
is the case. But if efficiently solved were asked and required of us
to list all 10^603 actual perfect
squares we would not have enough time. Another example is the Mersenne
Primes, in that the solutions are efficient
and the verification is efficient. Another example is the twin primes
in which there are more than 10^603 twin
primes between 0 and 10^1206, but we do not have the time to list all
of them, but we can check efficiently that such is true. So in
general, P=/NP, although there are cases in which they are equal, in
general they are not.
So the term "versus" is a curiosity, as to Finite versus Infinite is
the default zone of the P versus NP.
Archimedes Plutonium
> Few remarks:
> 1. "Time" in context of P and NP is a well defined mathematical concept. It counts steps that Turing machine makes in solving a problem. BTW, "machine" is also a defined concept: it is a discrete system of iterating a certain function. It might be confusing to people without CS background, but it is mostly standard process in math.
>
It is not confusing to a mathematician that CS people are trying to
conn the NP as a
mathematics problem when it is not. Until and unless the CS people
mathematically
define, solving, verification and time, they are whistling in the
wind.
And your notion of "time" intercepts with the well-defined concept of
sequence. So are we
going to say that "time" is a sequence? And how is time thence related
to physics "time".
As I said so often before, that the P versus NP problem is not a
mathematics problem because it fails on numerous terms of ill-
definition. You have to precisely define terms such as "time",
"efficiently solved", "efficiently verified" before you can claim it
is a
mathematical problem. If P versus NP is claimed to be a mathematics
problem, then
every problem of physics and engineering would be claimed as a
mathematics enterprize
for which clearly it is not.
> 2. "versus" is an old Latin word. And no, you in the "finite vs. infinite" do _not_ use it in the same sense as in "P vs. NP", because the relation there is pretty clear: finite is obviously _less than_ infinite.
>
You have not thought it out clear enough. The number 10^-604 is
smaller than 10^603
but is an infinite number.
Now if you restrict the topic of math to only Natural Numbers, then
all of these from 0 to 1 to
10^603 would allow for less than infinite. But in the larger set of
Algebraic Completeness
we need something like the concept of "versus" since division makes
the Algebraic
Completeness set from 10^-1206 to 10^1206, so that you have infinite
numbers scattered
in between finite numbers. And thus, plainly we need that concept of
"versus".
> 3. 10^1206+1 is a natural number.
Only for someone who cannot think outside of Old-Math box. The Peano
axioms are a victim of New-Math and seen as a half-baked axiomatic
set.
Now here I could use the help of LWalk in well-defining "versus" as a
new math concept.
I already mentioned some of its characteristics as being sort of
related to another item of
mathematics, yet possessing a distance apart. The concepts in math of
conjunction or "and" and the concept of "related to" are insufficient
to describe this new math concept of
"versus".
We see already the concept of "versus" used in Finite versus Infinite
and in P versus NP. But here is another use of "versuse" in that
Primes versus Composites.
Now we cannot use "function" to substitute for "versus", and
conjunction of "and" and "related to" are insufficient.
I believe this new math concept of "versus" is a Set theory concept.
And we can use the
Primes versus Composites as a template of what the concept covers.
Primes are related to
composites and vice versa (sorry for that pun). Primes are a disjoint
set from the composites, as well as Finites are a disjoint set from
Infinites, as we can see with Reals of
10^603 borderline of infinity that 10^-1206 and 10^1206 are infinites
yet some are smaller
than 10^603. And for algebraic completeness we also must include
negatives for subtraction. So they are disjoint sets, yet they have a
border in common. And they
are discrete, even the Natural and the Real forms of finite versus
infinite.
So the concept of "versus" as a mathematical term, is needed in New
Math, and it has Set
Theory origins, for it is seen as disjoint sets having some
relationship of one to the other.
Archimedes Plutonium
From: Archimedes Plutonium <plutonium.archime...@gmail.com>
Date: Wed, 2 Feb 2011 12:47:14 -0800 (PST)
Local: Wed, Feb 2 2011 2:47 pm
Subject: well defining a new term in mathematics -- "versus" #53
Correcting Math 3rd ed.
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Archimedes Plutonium
Archimedes Plutonium
of "versus" should be is
the concept of "compare" and mathematics already has a "comparative
analysis".
So when we use the term "versus" in mathematics such as these:
Primes versus Composites
P versus NP
Finite versus Infinite
That the "versus" term comes from Set theory and is a comparison of
one set to another set. As far as I can
see the above is a disjoint sets for primes versus composites and for
finite versus infinite.
So maybe mathematics already has a term of "compare" and that we do
not need another term of
"versus"
I think these sound just as good as the above:
Primes compared to Composites
P compared to NP
Finite compared to Infinite
We should not saddle mathematics, the science of precision with an
overburden of terms.
But we should have a bare minimum of precision terms. And this is
serendipity of the
discussion on P compared to NP in that such a problem has at least
three terms that are
nonmathematical 1) time 2) efficient solutions 3) efficient
verification, making the entire problem a piece of nonmath.
Veky
> On Feb 2, 2:22 am, Veky <ved...@gmail.com> wrote:
> > Few remarks:
> > 1. "Time" in context of P and NP is a well defined mathematical concept. It counts steps that Turing machine makes in solving a problem. BTW, "machine" is also a defined concept: it is a discrete system of iterating a certain function. It might be confusing to people without CS background, but it is mostly standard process in math.
> It is not confusing to a mathematician that CS people are trying to
> conn the NP as a
> mathematics problem when it is not. Until and unless the CS people
> mathematically
> define, solving, verification and time, they are whistling in the
> wind.
Yes, all those notions are precisely defined. Read Sipser, for example.
> And your notion of "time" intercepts with the well-defined concept of
> sequence. So are we
> going to say that "time" is a sequence?
Not exactly. "time", in context of CS, is a class of functions mapping size (number of symbols) of input to size (number of steps) of computation. You can look at it as a sequence, but not really in the sense you mean.
> And how is time thence related
> to physics "time".
Not very much. But it doesn't really matter. How is mathematical notion of "root" related to biological one? :-)
> As I said so often before, that the P versus NP problem is not a
> mathematics problem because it fails on numerous terms of ill-
> definition. You have to precisely define terms such as "time",
> "efficiently solved", "efficiently verified" before you can claim it
> is a
> mathematical problem.
Yes, again, all these things are defined. Although it might seem strange to you.
> If P versus NP is claimed to be a mathematics
> problem, then
> every problem of physics and engineering would be claimed as a
> mathematics enterprize
> for which clearly it is not.
No, of course not. "Time" in context of CS is in no way the _same_ thing as physical time. It is modelled after it, but they are not the same.
Is mathematical parabola the _same_ thing as a free fall? Of course not.
> > 2. "versus" is an old Latin word. And no, you in the "finite vs. infinite" do _not_ use it in the same sense as in "P vs. NP", because the relation there is pretty clear: finite is obviously _less than_ infinite.
> You have not thought it out clear enough. The number 10^-604 is
> smaller than 10^603
> but is an infinite number.
I thought we were talking about natural numbers, of course.
> Now if you restrict the topic of math to only Natural Numbers, then
> all of these from 0 to 1 to
> 10^603 would allow for less than infinite. But in the larger set of
> Algebraic Completeness
> we need something like the concept of "versus" since division makes
> the Algebraic
> Completeness set from 10^-1206 to 10^1206, so that you have infinite
> numbers scattered
> in between finite numbers. And thus, plainly we need that concept of
> "versus".
You are confusing ordering numbers by their standard order, and by cardinality. 1/2 is also infinite in cardinality.
> > 3. 10^1206+1 is a natural number.
>
> Only for someone who cannot think outside of Old-Math box.
I'll happily include myself in that group. :-)
> Primes are related to
> composites and vice versa (sorry for that pun).
It's _not_ a pun, but a necessity. Once you understand that, you'll see what I was talking about above.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
(snipped)
> And this is
> serendipity of the
> discussion on P compared to NP in that such a problem has at least
> three terms that are
> nonmathematical 1) time 2) efficient solutions 3) efficient
> verification, making the entire problem a piece of nonmath.
>
On Feb 3, 4:24 am, Veky <ved...@gmail.com> wrote:
> On Wednesday, February 2, 2011 9:47:14 PM UTC+1, Archimedes Plutonium wrote:
> > On Feb 2, 2:22 am, Veky <ved...@gmail.com> wrote:
> > > Few remarks:
> > > 1. "Time" in context of P and NP is a well defined mathematical concept. It counts steps that Turing machine makes in solving a problem. BTW, "machine" is also a defined concept: it is a discrete system of iterating a certain function. It might be confusing to people without CS background, but it is mostly standard process in math.
> > It is not confusing to a mathematician that CS people are trying to
> > conn the NP as a
> > mathematics problem when it is not. Until and unless the CS people
> > mathematically
> > define, solving, verification and time, they are whistling in the
> > wind.
>
> Yes, all those notions are precisely defined. Read Sipser, for example.
>
> > And your notion of "time" intercepts with the well-defined concept of
> > sequence. So are we
> > going to say that "time" is a sequence?
>
> Not exactly. "time", in context of CS, is a class of functions mapping size (number of symbols) of input to size (number of steps) of computation. You can look at it as a sequence, but not really in the sense you mean.
>
> > And how is time thence related
> > to physics "time".
>
> Not very much. But it doesn't really matter. How is mathematical notion of "root" related to biological one? :-)
>
> > As I said so often before, that the P versus NP problem is not a
> > mathematics problem because it fails on numerous terms of ill-
> > definition. You have to precisely define terms such as "time",
> > "efficiently solved", "efficiently verified" before you can claim it
> > is a
> > mathematical problem.
>
> Yes, again, all these things are defined. Although it might seem strange to you.
>
> > If P versus NP is claimed to be a mathematics
> > problem, then
> > every problem of physics and engineering would be claimed as a
> > mathematics enterprize
> > for which clearly it is not.
>
> No, of course not. "Time" in context of CS is in no way the _same_ thing as physical time. It is modelled after it, but they are not the same.
>
> Is mathematical parabola the _same_ thing as a free fall? Of course not.
>
> > > 2. "versus" is an old Latin word. And no, you in the "finite vs. infinite" do _not_ use it in the same sense as in "P vs. NP", because the relation there is pretty clear: finite is obviously _less than_ infinite.
> > You have not thought it out clear enough. The number 10^-604 is
> > smaller than 10^603
> > but is an infinite number.
>
> I thought we were talking about natural numbers, of course.
>
> > Now if you restrict the topic of math to only Natural Numbers, then
> > all of these from 0 to 1 to
> > 10^603 would allow for less than infinite. But in the larger set of
> > Algebraic Completeness
> > we need something like the concept of "versus" since division makes
> > the Algebraic
> > Completeness set from 10^-1206 to 10^1206, so that you have infinite
> > numbers scattered
> > in between finite numbers. And thus, plainly we need that concept of
> > "versus".
>
> You are confusing ordering numbers by their standard order, and by cardinality. 1/2 is also infinite in cardinality.
>
> > > 3. 10^1206+1 is a natural number.
>
> > Only for someone who cannot think outside of Old-Math box.
>
> I'll happily include myself in that group. :-)
>
> > Primes are related to
> > composites and vice versa (sorry for that pun).
>
> It's _not_ a pun, but a necessity. Once you understand that, you'll see what I was talking about above.
The above is a person enamored by Computer Science (CS) who is not a
mathematician, and who hates to admit that the P versus NP is not a
problem of mathematics.
There is a huge desire by nonmathematicians to include their CS
imaginations as a mathematical proposition, as a piece of mathematics.
And the P versus NP is one of those
dreadful cases of a piece of nonmath trying to be claimed as math.
Before the P versus NP is claimed as mathematical it needs to list
precision definitions of the above shortfalls
(a) time precisely defined relative math's sequence
(b) efficient solution precisely defined
(c) efficient verification precisely defined
Mathematics has another example of a problem that was never given
precision definitions
to be a math problem in the first place. I am talking about the 4
Color Mapping with its absurd notion that countries can exist yet not
have borders between them-- borderless countries is an extreme
absurdity. And the 4 Color Mapping when it concedes to having borders,
is easily proven in a paragraph by the Jordan Curve Theorem. I want no
further discussion of this now, since I devote a chapter to 4 Color
Mapping in this book.
The reason I started this book with the Euclid Infinitude of Primes
proof is to show to the world that most mathematicians, 90%
approximately, could not even do a valid Euclid Infinitude of Primes
without tripping all over themselves with mistakes galore in their
attempts. That most could not even recognize a valid reductio ad
absurdum.
And then, to think that Computer Science people without precision
definitions of time, efficient solution, efficient verification, come
running over to the science of mathematics
and hand off a silly proposition of P versus NP as mathematics. Give
us a break. Will CS
people next want a proof that "fluffy the cat is part rat?"
Veky
> There is a huge desire by nonmathematicians to include their CS
> imaginations as a mathematical proposition, as a piece of mathematics.
> And the P versus NP is one of those
> dreadful cases of a piece of nonmath trying to be claimed as math.
>
> Before the P versus NP is claimed as mathematical it needs to list
> precision definitions of the above shortfalls
>
> (a) time precisely defined relative math's sequence
> (b) efficient solution precisely defined
> (c) efficient verification precisely defined
All these have been done. You haven't read Sipser, have you?
Ok, I'll make it easier for you. Do you accept the notions
* Turing machine
* sequence
* polynomial (in one variable)
as mathematical entities?
If you do, I can define (a), (b) and (c) for you.
Archimedes Plutonium
it is
not a mathematics problem at all for it lacks precision definitions of
its
concepts involved. It has no precision definition of (a) time nor (b)
efficient
solution nor (c) efficient verification, plus other concepts involved.
Everyone can understand that a new science like Computer Science
started in
the mid-1900s was anxious to become more respectable, and so was
anxious to
try to get in the same league of science as say mathematics. So anyone
can
understand that a new science of CS would try its very best to pose a
question
that would look as though it was a mathematics question when in fact
it was
a cloudy and obfuscated question of electrical engineering and
physics. But there
were enough so called "suckers in math" to pass the P versus NP as a
honorable
question of mathematics when in fact it was never that. One must not
forget that
the mathematics community could not even fix the Euclid Infinitude of
Primes proof
of a valid Direct method from a valid Indirect Method and where a 90%
of math professors in
books could not show a error free proof. Keeping that in mind, here we
have a
CS question where the mathematics community thinks the P versus NP is
a actual math problem. So can we expect those mathematics professors
who could not even do a valid
Euclid Infinitude of Primes proof, can we expect them to get something
like P versus NP
with so many new subtle terms such as "time", "efficient
verification", can we really
expect that mathematics community which 90% could not even do a simple
proof of Euclid
Infinitude of Primes validly, and have them address something as
complex as P versus NP.
Can we expect grade-school children to build a electric nuclear power
plant, is of the
same category.
What I want to accomplish in this post, is to begin to make the P
versus NP problem of
Computer Science into a truly mathematical problem. I only want to
start that, for I plan
to include P versus NP as a full chapter in this textbook of
Correcting Math.
I have often asked the question in sci.math, beginning in the 1990s
for someone to deliver a mathematical equivalent statement of the P
versus NP problem. We only have to look at
the Riemann Hypothesis to see a myriad equivalent statements, and even
the 4 Color Mapping has some equivalent statements. But when we focus
on the P versus NP,there are no equivalent statements, only the
couched and entrenched terminology of computer science statement of P
versus NP. The reason that there are no equivalent statements is
because P versus NP was never a mathematics problem at all. I cannot
run to a math department and expect them to solve the statement: "life
is in heirarchies of stability" because that is not mathematics even
though it has some terms that may look mathematical.
So here I want to start a statement formulation that perhaps is at the
heart of what the P versus NP was trying to question and to solve.
Notice in mathematics proper that Euler had found the formula n^2 + n
+41 for n<40 predicted all the primes in that small interval. The
primes have no overall formula that
predicts every prime. So one can see already some relationship to P
versus NP of its
"time" "efficient solving" and "efficient verification". Let me
continue.
So now it is reasonable to think that we can have smallish formulas
all along the string of
primes out to 10^603 with a smallish formula that predicts every prime
in that smallish
interval just as the n^2 + n +41 did.
So here we can formulate a Mathematical proposition that is probable
what the P versus NP
was trying to grapple with. We can formulate a statement that all the
primes from 0 to
10^603 have these smallish intervals given by a smallish formula.
Now that is only a start.
There is another nagging question that comes to my mind. I am looking
for a connection of the point of infinity, the borderline of infinity
as given by pi as the first three zeroes
in a row at 10^-603. I want a connection of three zeroes in a row with
that of the primes.
We know that pi is related to the primes, so that there must be a
prime characteristic at 10^603 that ties into directly with the three
zeroes in a row. It maybe the feature that these formulas like Euler's
of n^2 +n +41 stop and exhaust themselves for the first time at
10^603. So that before 10^603, the primes yield these smallish
formulas but halt at 10^603.
And this is not a P versus NP question,but my own personal question of
the connection of pi
at 10^-603 with primes at 10^603.
Archimedes Plutonium
question P = NP? asks:
Suppose that solutions to a problem can be verified quickly. Then, can
the solutions themselves also be computed quickly?"
Now what I propose to making that question of Computer Science into a
mathematics
question or conjecture is to substitute the primes versus composites
rather than P versus
NP. And from Euler we know that a stretch interval of primes can be
exactly determined by a
interval formula such as N^2 + N +41 for N<4 for the primes 43, 47,
53. Now that is only three consecutive primes, but imagine a huge
number of formulas like that to pinpoint every prime in its succession
and we have a cutoff point such as 10^603. Now the world does not
have a computer that can list in order all the primes to 10^603 so the
P versus NP problem translated into mathematics is not a question of
time for solution computing nor a question of time of verification.
Rather instead, a mathematical translation of P versus NP would ask
whether there exists a unending string of these Euler type formulas
that exactly predicts all the primes in order? If such a existence is
true then P=NP, if such an existence is not true then P=/NP.
So to make the P versus NP a truly mathematical question entails
getting rid of all
obfuse concepts that are nonmathematical such as time, efficient
solving, and efficient verification and substituting Primes able to be
exactly ordered by snippets of formulas.
It is already known in mathematics that no one formula can predict all
the primes. But is there snippets of formulas that exactly predict the
primes of a snippet interval. Such a question is a mathematical
equivalent question of the CS problem of P versus NP.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
> Repeating the quote of the P versus NP problem "In essence, the
> question P = NP? asks:
> Suppose that solutions to a problem can be verified quickly. Then, can
> the solutions themselves also be computed quickly?"
>
> Now what I propose to making that question of Computer Science into a
> mathematics
> question or conjecture is to substitute the primes versus composites
> rather than P versus
> NP. And from Euler we know that a stretch interval of primes can be
> exactly determined by a
> interval formula such as N^2 + N +41 for N<4 for the primes 43, 47,
> 53. Now that is only three consecutive primes, but imagine a huge
Correction there, for N= 0 gives the prime 41 so that snippet formula
gives
four consecutive primes in a row.
> number of formulas like that to pinpoint every prime in its succession
> and we have a cutoff point such as 10^603. Now the world does not
> have a computer that can list in order all the primes to 10^603 so the
> P versus NP problem translated into mathematics is not a question of
> time for solution computing nor a question of time of verification.
> Rather instead, a mathematical translation of P versus NP would ask
> whether there exists a unending string of these Euler type formulas
> that exactly predicts all the primes in order? If such a existence is
> true then P=NP, if such an existence is not true then P=/NP.
>
> So to make the P versus NP a truly mathematical question entails
> getting rid of all
> obfuse concepts that are nonmathematical such as time, efficient
> solving, and efficient verification and substituting Primes able to be
> exactly ordered by snippets of formulas.
>
> It is already known in mathematics that no one formula can predict all
> the primes. But is there snippets of formulas that exactly predict the
> primes of a snippet interval. Such a question is a mathematical
> equivalent question of the CS problem of P versus NP.
>
Now if I had to hazard a guess as to the outcome of the above, I would
say that
some point of the Natural Numbers is a breakdown point, where we have
these
"snippet formulas" able to predict all the primes in succession up
until we reach
that breakdown point. And I would hazard to guess that the breakdown
point is the
prime in the neighborhood of 10^603 having some of its digits as
215215--- pattern.
This would be highly gratifying to me, because I am searching for the
connection between
primes and the digits of pi where at 10^-603 pi has three zero digits
in a row for the first time and where a finite perimeter cannot be
matched by a finite circumference. So if
the primes had snippet-formulas existing all the way up to 10^603 but
have a breakdown at
10^603 would be a vindication to me of the deep intimate connection of
pi and the primes.
Archimedes Plutonium
You know, a lot of people reckon that intuition is like instinct and a
perfect example
is this desire by me to find what the connection is of primes and pi.
So I have
pi with three zero digits in a row at 10^-603 which prevents a finite
perimeter to be
matched, for the first time to a finite circumference. It has 603
digits and its arrangement
has a pattern of 215215----. So I want to know badly, what the primes
relationship is
at 10^603? If pi fails to deliver a finite circumference, because of
those three zero digits in a
row, and since pi is intimately related to the primes, just look at pi
= 4 - 4/3 + 4/5 - 4/7+. . .
So how do the primes act up at 10^603?
And the answer comes from P versus NP, which is changed to Primes
versus Composites.
Using Euler with his famous algorithm N^2 +N+41 for N<4 delivers four
primes exactly in a row
of 41, 43, 47, 53. Now set up these "snippet formulas" all along the
Primes. N^2 + N+5 for N<3
delivers to us 5, 7, 11 where a bare minimum is two primes in
succession with first case starting at
5. And we can vary the snippet-formula where we are not stuck to one
formula.
So what is the connection of primes to pi at 10^-603 with its three
zeroes in a row that forces Finite
to breakdown with Infinity? The connection is that primes at 10^603
fail to have a snippet formula
that connects those three primes in succession at 10^603.
So, just as there is a finite perimeter of a square that is not
matched by a finite circle circumference
at 10^603, also, for primes there is no snippet formula at 10^603 that
bridges three primes there.
Veky
> bridges three primes there.
10^603+103+1886N-596N^2.
Archimedes Plutonium
out the primes. But now there is
the Net of AP, which is a reversal of a sieve. Here we use primes to
bridge other primes. And the bridge
is the formula N^2 +N +prime
We exclude the primes 2 and 3 and start with 5
5 7 11 13 17 19 23
29
31 37 41
43 47 53 59 61 67
71
73 79 83 89
97 101 103 107 109
113
127 131 137 139
149 151 157 163 167
173
179 181 191 193
197 199 211 223 227
N^2 +N+5 yields 5,7,11,17
N^2 +N+11 yields 11,13,17,23,31,41,53,67,83,101
N^2 +N+17 yields 17,19,23,29,37,47,59,73,89,107,127,149
N^2 +N+41 yields 41,43,47,53,61,71,83,97, etc
Now there is one problem so far; I cannot seem to reach 79. Will try
later tonight. Maybe I should use subtraction also in the formula.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
> There is the Sieve of Eratosthenes, which is famously known to distill
> out the primes. But now there is
> the Net of AP, which is a reversal of a sieve. Here we use primes to
> bridge other primes. And the bridge
> is the formula N^2 +N +prime
>
> We exclude the primes 2 and 3 and start with 5
> 5 7 11 13 17 19 23 29 31 37 41
> 43 47 53 59 61 67 71 73 79 83 89
> 97 101 103 107 109 113 127 131 137 139
> 149 151 157 163 167 173 179 181 191 193
> 197 199 211 223 227
> N^2 +N+5 yields 5,7,11,17
> N^2 +N+11 yields 11,13,17,23,31,41,53,67,83,101
> N^2 +N+17 yields 17,19,23,29,37,47,59,73,89,107,127,149
> N^2 +N+41 yields 41,43,47,53,61,71,83,97, etc
>
> Now there is one problem so far; I cannot seem to reach 79. Will try
> later tonight. Maybe I should use subtraction also in the formula.
>
Now I could reach 79 from 59 via
N^2 +N+59 yields 59,61,65,71,79
But that lets in a composite into the bridge of 65.
Since the Wallis formula of pi to primes is of a alternating add
subtract, maybe I can resolve it that way.
Archimedes Plutonium
Now let me not mitigate the problem I am having in making that net
smooth
using the N^2 + N +prime as the bridge.
But there is one way of making it smooth, and perhaps it is the reason
that
no formula covers all the primes. So with this patch, call it the AP-
patch,
is to make believe that odd numbers ending in a "5" digit such as 65
are permissible.
The digit as the ending digit of odd numbers has always been special
in mathematics
as the only odd number digit that cannot be prime beyond 5 itself.
So this AP-patch when stuck to formulas, may provide a formula that
encompasses or
nets every prime number by a formula, provided we use that patch.
Let me check to see if that is the case.
Archimedes Plutonium
patch."
If I include N^2 +N+7 , N>7 yields 79,97,107,139 I end up getting 79
So the N starting with 0 then 1, then 2 etc, was too restrictive. If I
open it up to
N^2 +N+prime to an N>q,then I perhaps can retrieve every prime from
these formulas.
2 3 5 7 11 13 17 19 23 29
31 37 41 43 47 53 59 61 67
71
73 79 83 89 97 101 103 107 109
113
127 131 137 139 149 151 157 163 167
173
179 181 191 193 197 199 211 223 227
N^2 +N+5 yields 5,7,11,17
N^2 +N+7 , N>7 yields 79,97,107,139
N^2 +N+11 yields 11,13,17,23,31,41,53,67,83,101
N^2 +N+17 yields 17,19,23,29,37,47,59,73,89,107,127,149
N^2 +N+41 yields 41,43,47,53,61,71,83,97, etc
Archimedes Plutonium
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
> There is a better means, rather than the "ending digit of 5 as a
> patch."
>
> If I include N^2 +N+7 , N>7 yields 79,97,107,139 I end up getting 79
>
> So the N starting with 0 then 1, then 2 etc, was too restrictive. If I
> open it up to
> N^2 +N+prime to an N>q,then I perhaps can retrieve every prime from
> these formulas.
>
> 2 3 5 7 11 13 17 19 23 29
> 31 37 41 43 47 53 59 61 67 71
> 73 79 83 89 97 101 103 107 109 113
> 127 131 137 139 149 151 157 163 167 173
> 179 181 191 193 197 199 211 223 227
> N^2 +N+5 yields 5,7,11,17
> N^2 +N+7 , N>7 yields 79,97,107,139
> N^2 +N+11 yields 11,13,17,23,31,41,53,67,83,101
> N^2 +N+17 yields 17,19,23,29,37,47,59,73,89,107,127,149
> N^2 +N+41 yields 41,43,47,53,61,71,83,97, etc
>
Alright, I am confident the formula N^2 +N+prime can bridge all the
primes and
let me call it the Net of AP, a reverse of the Sieve of Eratosthenes.
Here I want
to fetch every prime number with one formula that bridges all the
primes.
But here is where the formula becomes entwined with the P versus NP
question. There is
no way of verifying all the primes from 5 to the last primes of the
neighborhood of
10^1206. There is no way of any computer of crunching through all
those primes to see
if the formula snippets as bridges spans and covers every prime.
And my conjecture that since pi is connected to primes and since pi
has three zero digits in
a row at 10^-603, the conjecture that a breakdown of that formula N^2+N
+prime has a breakdown
at 10^603 region of primes.
So here I have a junction or juncture point of primes, pi, and the P
versus NP.
Archimedes Plutonium
41 43 47 53 59 61 67 71 73 79
83 89 97 101 103 107 109 113 127 131
137 139 149 151 157 163 167 173 179 181
191 193 197 199 211 223 227
N^2 +N+5 yields 5,7,11,17
N^2 +N+7 , N>7 yields 79,97,107,139
N^2 +N+11 yields 11,13,17,23,31,41,53,67,83,101
N^2 +N+17 yields 17,19,23,29,37,47,59,73,89,107,127,149
N^2 +N+41 yields 41,43,47,53,61,71,83,97, etc
Now one must ask, if that snippet formula N^2+N+prime bridges all the
primes (except I hope a breakdown at 10^603).
Then the next question would have to be why such a formula as
representative of all the primes, or nearly all the primes? What is so
special about N^2 +N+ prime?
I would have to say the specialness of that formula is that it is a
inverse of the idea that between N and 2N exists a prime and so
between N^2 and N exists a prime. And so the formula-snippets captures
all those primes.
Archimedes Plutonium
formula
N^2 + N +prime, especially with capturing 79 and I was fretting and
perturbed until I realized I need not be so restrictive with starting
out
always with 0,1,2,3,.... but could start out with say 8,9,10 as in
N^2 +N+7 , N>7 yields 79,97,107,139
So everything worked out, that these snippet-formulas, as I like to
call them
form a bridge and capture all the primes starting at 5 in sequences
that are only
primes as the below notes.
5 7 11 13 17 19 23
29
31 37
41 43 47 53 59 61 67
71
73 79
83 89 97 101 103 107 109
113
127 131
137 139 149 151 157 163 167
173
179
181
191 193 197 199 211 223 227
N^2 +N+5 yields 5,7,11,17
N^2 +N+7 , N>7 yields 79,97,107,139
N^2 +N+11 yields 11,13,17,23,31,41,53,67,83,101
N^2 +N+17 yields 17,19,23,29,37,47,59,73,89,107,127,149
N^2 +N+41 yields 41,43,47,53,61,71,83,97, etc
Now one must ask, if that snippet formula N^2+N+prime bridges all the
primes (except I hope a breakdown at 10^603).
Then the next question would have to be why such a formula as
representative of all the primes, or nearly all the primes? What is
so
special about N^2 +N+ prime?
I would have to say the specialness of that formula is that it is a
inverse of the idea that between N and 2N exists a prime and so
between N^2 and N exists a prime. And so the formula-snippets
captures
all those primes.
Now we all have been told and memorized the message that the Primes
have no formula that
delivers all the primes. But no mathematician has ever really delved
into the concept
of snippet-formulas that bridges all the primes.
And that general snippet formula of N^2 +N +prime is a function that
is similar to the parabola of mathematics as F(x) = ax^2 + bx + c
So, are all the primes located on a parabolic function? Considering
that all the primes can be retrieved via these snippet formula
bridges.
We all know the famous Sieve of Eratosthenes that filters out the
composites and ends up with only primes. But consider the opposite of
a sieve as a production-generator where
you put numbers into the generator and out comes the primes.
Now I ventured into this topic by happenstance, for I was talking
about the P versus NP problem and also wondering since the borderline
of infinity is 10^603 because pi has three
zero digits in a row, was wondering if something funny or special
happened with the primes
at 10^603. Primes are related to pi, so if pi acts up strangely at
10^-603 then we can expect some strange behaviour of primes at 10^603.
Since Circumferencing finite circles with matching of finite square
perimeters has a breakdown at 10^603, can we expect some breakdown of
the primes at 10^603? What I mean is, can those snippet-formulas have
a breakdown at 10^603 so that there is a unique prime in which for the
first time, that parabolic prime formulas does have a breakdown and
unable to be bridged by those snippet
formulas. I almost had a breakdown of the prime 79 today, but was able
to overcome that
difficulty.
And here is where the P versus NP comes into play, because I can see
no way in which anyone
can verify or solve that question of whether all primes are bridged
until they reach 10^603.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
> Earlier today I had a problem of capturing all the primes with the
> formula
> N^2 + N +prime, especially with capturing 79 and I was fretting and
> perturbed until I realized I need not be so restrictive with starting
> out
> always with 0,1,2,3,.... but could start out with say 8,9,10 as in
>
> N^2 +N+7 , N>7 yields 79,97,107,139
>
Sorry, I made a mistake there. That should have been 117 which is not
a prime for N=10. So that the sequence of N^2 +N+7, N>7 is a snippet-
formula
of a bridge that contains just two primes. But an important bridge.
And since the formula N^2+N +prime is capable of covering all the
primes, unless
we use a P versus NP to verify. That we have a Primes versus
Composites
as a substitute for the P versus NP conjecture.
Now in a future edition of this book, I shall make those separate
chapters.
Now I should mention something about the fact that the three zeroes in
a row in pi at 10^-603
cause or force our first finite perimeter of a square to not be
matched by a finite circle circumference
and since pi and primes are related via the Wallis formula pi = 4 -
4/3 +4/5 - 4/7+... That we can
ask what is the behaviour of the primes at 10^603? Do the primes act
badly at 10^603 and have
some sort of finite to infinite breakdown? So if perimeter and
circumference have a radical
change at 10^603 then we can expect the primes to have some sort of
change of characteristic.
And what I conjecture is that the snippet-formula for all the primes
as N^2+N+prime is a solid bridge
that includes all the primes from 5, all the way up to 10^603 where in
fact the primes have a breakdown
of that formula where it fails to bridge a prime in that 10^603
region. So here is a beautiful mingling of
mathematical topics of the parabola formula and whether pi is related
to the parabola, and whether P versus NP
is true or false and can be substituted by Primes versus Composites.
So we have a lot going on here.
And we may even have the Riemann Hypothesis involved, for RH was in
the complex-plane asking if all
the primes are in the straight line of the 1/2 Real. But here we are
asking if all the primes lie on a Parabolic
curve composed of snippet formulas. And if the primes have a
counterexample at 10^603, would say that
RH is false.
Archimedes Plutonium
So the snippet formulas that bridges all the primes, at least we
conjecture all the primes
is N^2 +N +prime which is a parabola formula of F(x) = ax^2 + bx + c
So I went around checking as to the relationship of pi to the parabola
and found that the
trapezoid rule and parabola are techniques to fetch the digits of pi.
Also, that the parabola is the limit of sequences of ellipses with a
focal point at infinity.
Infinity here would be 10^603. And we know that ellipses are just
simply squashed
circles. So this looks promising.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
> So the snippet formulas that bridges all the primes, at least we
> conjecture all the primes
> is N^2 +N +prime which is a parabola formula of F(x) = ax^2 + bx + c
>
> So I went around checking as to the relationship of pi to the parabola
> and found that the
> trapezoid rule and parabola are techniques to fetch the digits of pi.
>
> Also, that the parabola is the limit of sequences of ellipses with a
> focal point at infinity.
> Infinity here would be 10^603. And we know that ellipses are just
> simply squashed
> circles. So this looks promising.
>
Yes indeed, this is coming together pretty. If given the formulas of
conic sections
which would one guess would be the formula that patterns the primes?
Desargues found that the ellipse has zero points at infinity, and the
parabola has one point at infinity and the
hyperbola has two points at infinity.
So the formula to pick that patterns the primes the best is the
parabola Y = ax^2 +bx + c.
Now I do not know if Euler found his N^2 +N+prime from that of the
parabola? Probably not. He probably found it from experimenting with
formulas.
But what I want to find is that pi is related to primes and pi has
three zero digits in a row at 10^-603 that forms a border of finite
with infinity. So that the Desargues's point at infinity for the
parabola would be 10^603. Now I need to find out if that number 10^603
or its square of 10^1206 as the algebraic completeness, would fit
perfectly with some other facts of mathematics. Is the 10^603 and
10^1206 forming the two "points of infinity of Desargues" for the
hyperbola? Very very interesting.
Archimedes Plutonium
Back in NOV2010
Enrico wrote:
> Begin Divisibility by 120 search:
> D I Mod
> 0 246 0
> 314159265358979323846264338327950288419716939937510582097494459230781640628 620899862803482534211706798214808651328230664709384460955058223172535940812 848111745028410270193852110555964462294895493038196442881097566593344612847 564823378678316527120
> 0 602 0
> 314159265358979323846264338327950288419716939937510582097494459230781640628 620899862803482534211706798214808651328230664709384460955058223172535940812 848111745028410270193852110555964462294895493038196442881097566593344612847 564823378678316527120190914564856692346034861045432664821339360726024914127 372458700660631558817488152092096282925409171536436789259036001133053054882 046652138414695194151160943305727036575959195309218611738193261179310511854 807446237996274956735188575272489122793818301194912983367336244065664308602 139494639522473719070217986094370277053921717629317675238467481846766940513 20
> 0 603 0
> 314159265358979323846264338327950288419716939937510582097494459230781640628 620899862803482534211706798214808651328230664709384460955058223172535940812 848111745028410270193852110555964462294895493038196442881097566593344612847 564823378678316527120190914564856692346034861045432664821339360726024914127 372458700660631558817488152092096282925409171536436789259036001133053054882 046652138414695194151160943305727036575959195309218611738193261179310511854 807446237996274956735188575272489122793818301194912983367336244065664308602 139494639522473719070217986094370277053921717629317675238467481846766940513 200
> 0 604 0
> 314159265358979323846264338327950288419716939937510582097494459230781640628 620899862803482534211706798214808651328230664709384460955058223172535940812 848111745028410270193852110555964462294895493038196442881097566593344612847 564823378678316527120190914564856692346034861045432664821339360726024914127 372458700660631558817488152092096282925409171536436789259036001133053054882 046652138414695194151160943305727036575959195309218611738193261179310511854 807446237996274956735188575272489122793818301194912983367336244065664308602 139494639522473719070217986094370277053921717629317675238467481846766940513 2000
That was an important message of the divisibility of pi digits, as
evenly divisible
by 120, begot from regular polygons and regular polyhedra of
3,4,5,6,8,12,20.
The importance of that even divisiblity is that it eliminates the pi
digits of
10^-308 where pi has its first two zero digits in a row as being the
border of
infinity. Mind you, computers have not yet proven that 10^-603 is the
first
breakdown of at least one finite perimeter unmatched by a finite
circumference,
but the proof that pi is not evenly divisible by 120 at the 10^-308 is
strong
indication that 10^308 is not the borderline of infinity but rather
10^603 is
that borderline.
So why bring up that topic here? Well, because if a parabola curve
describes
all the primes via N^2 +N+prime as snippet bridges, and via Desargues
projective
geometry there is one point of infinity for a parabola, implies that
10^603
is the singularity point.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
(snipped)
> So why bring up that topic here? Well, because if a parabola curve
> describes
> all the primes via N^2 +N+prime as snippet bridges, and via Desargues
> projective
> geometry there is one point of infinity for a parabola, implies that
> 10^603
> is the singularity point.
>
Now it is a good bet that the Finite versus Infinite breakdown at
10^603 involves a number with
602 digits, since its perimeter is finite for the 10^603 B matrix, and
that when multiplied by 4 for
the perimeter would be less than 10^603 itself-- hence a finite
perimeter, and unmatched by a
circle circumference. And from small B matrices of 100, 10^3, 10^4,
that number has a repeating
pattern of 215215----
215.50 for 10^2 B matrix
2152.050 for 10^3 B matrix
21521.5000 for 10^4 B matrix
215215.21500 for 10^5 B matrix
So we have this huge number with 602 digits with a digit pattern of
215215----
And we have those Parabolic Prime Formulas of N^2 +N+prime bridging
all the way up to
10^603, and as Desargues projective geometry states, that the parabola
has a singularity
point at infinity.
Would that singularity point be a prime number? I would guess so if we
stripped away the decimal
point, although the above would not be primes since they end in "5".
I was looking for a Fibonacci prime with 602 digits.
Archimedes Plutonium
Now there is no force of logic to force that point of infinity of the
parabola of formulas
N^2 +N+prime which is a parabolic curve of primes, to force that
"point of infinity"
to be a Fibonacci prime.
Now here is a mightly good website discussing Fibonacci primes:
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html
Fn2971
35710356064...16229 which has 621 digits
Fn5387
29304412869...55833 which has 1126 digits
Since pi has three zero digits in a row at 10^-603 it forces a finite
perimeter to not be matched
by a finite circumference and thus a borderline of Infinity. So, does
that mean that a Fibonacci
prime at 10^603 or its algebraic completeness at 10^1206 is a prime of
that Parabola of Primes
that is Fibonacci prime?
I happen to think so.
The above website of their listing of Fibonacci primes is not an all
inclusive listing. I suspect there
are quite a few Fibonacci primes which have exactly 603 digits and
quite a few that have exactly
1206 digits. I suspect that the 1206 digit Fibonacci primes, that one
of them looks like this
in its digit arrangement 215215---------
Keep in mind that we disregard the decimal point in the 10^603 B
matrix to look for that breakdown
point and ask whether it is a Fibonacci prime.
Archimedes Plutonium
On Feb 7, 2:34 pm, Archimedes Plutonium
>
> Fn2971
> 35710356064...16229 which has 621 digits
>
> Fn5387
> 29304412869...55833 which has 1126 digits
>
> Since pi has three zero digits in a row at 10^-603 it forces a finite
> perimeter to not be matched
> by a finite circumference and thus a borderline of Infinity. So, does
> that mean that a Fibonacci
> prime at 10^603 or its algebraic completeness at 10^1206 is a prime of
> that Parabola of Primes
> that is Fibonacci prime?
>
> I happen to think so.
>
> The above website of their listing of Fibonacci primes is not an all
> inclusive listing. I suspect there
> are quite a few Fibonacci primes which have exactly 603 digits and
> quite a few that have exactly
> 1206 digits. I suspect that the 1206 digit Fibonacci primes, that one
> of them looks like this
> in its digit arrangement 215215---------
>
> Keep in mind that we disregard the decimal point in the 10^603 B
> matrix to look for that breakdown
> point and ask whether it is a Fibonacci prime.
>
Now a nice model of this Desargues singularity point at infinity for
the parabola is described in the:
http://www.physicsforums.com/archive/index.php/t-25487.html
matt grime
May13-04, 06:13 AM
The idea he's getting at is that they 'meet' at infinity: there is a
model for the plane that adds a 'point at infinity', imagine the plane
in space, and a ball such that the plane cuts through the equator.
There is a projection map from the north pole to the plane obtained by
taking lines from the north pole - these pass through exactly one
point in the plane and one point in the sphere. so you can identify
the plane with the surface of the sphere minus the north pole. we call
teh north pole the point at infinity, as it has the property that
every curve in the plane that 'goes to infinity' tends towards the
north pole on the sphere.
HallsofIvy
May13-04, 06:30 AM
FIRST- "infinity" is not a single, well-defined point. Standard
geometry does not include a "point at infinity" but there are several
models that do. How you think about infinity depends upon which model
you use- matt grime gave the most common 2-dimensional model.
What your teacher was thinking about is probably this:
Start with an ellipse with one focus at (0,0), the other on the y-axis
(at (0,y), say). Imagine "stretching" the ellipse so that, while the
first focus remains at (0,0), the second focus moves along the y-axis
with y getting larger and larger. You will find that the eccentricity
of the ellipse increases- getting closer and closer to 1.
If you imagine stretching that ellipse out "to infinity"- so that the
second focus "goes to infinity", then the eccentricity goes to 1
(technically, "in the limit") so that the ellipse "becomes" a
parabola. That is the sense in which a parabola "closes at infinity"-
a parabola is like an infinitely long ellipse.
Of course, if stretch "to infinity and beyond" (!!) the eccentricity
becomes larger than 1- a hyperbola. The focus that went out to
infinity along the positive y axis now reappears on the negative y
axis! That happens in precisely the same sense that the hyperbola y= 1/
x has y going to infinity as x approaches 0 (from above) and then
reappears with negative y on the other side of 0.
In order to make that precise, you have to use the "spherical" model
for the plane that matt grime talked about.
--- end quoting http://www.physicsforums.com/archive/index.php/t-25487.html
---
What I like to point out is that 10^603 is taken as that singularity
point of infinity for
the parabola but that 10^1206 which heretofor I have called the
algebraic completion
is now seen geometrically as the hyperbola continuation of the
parabola. So in a sense
Algebraic Completeness of its operations is seen in terms of geometry
as the completion of ellipses plus parabolas into that of hyperbolas.
So we have a beauty of unification of algebra, geometry.
Archimedes Plutonium
http://www.iw.net/~a_plutonium
Archimedes Plutonium
to ask where pi and e are primes
in the 10^603 region.
Pi = 3.
1415926535 8979323846 2643383279 5028841971 6939937510 : 50
5820974944 5923078164 0628620899 8628034825 3421170679 : 100
8214808651 3282306647 0938446095 5058223172 5359408128 : 150
4811174502 8410270193 8521105559 6446229489 5493038196 : 200
4428810975 6659334461 2847564823 3786783165 2712019091 : 250
4564856692 3460348610 4543266482 1339360726 0249141273 : 300
7245870066 0631558817 4881520920 9628292540 9171536436 : 350
7892590360 0113305305 4882046652 1384146951 9415116094 : 400
3305727036 5759591953 0921861173 8193261179 3105118548 : 450
0744623799 6274956735 1885752724 8912279381 8301194912 : 500
9833673362 4406566430 8602139494 6395224737 1907021798 : 550
6094370277 0539217176 2931767523 8467481846 7669405132 : 600
0005681271 4526356082 7785771342 7577896091 7363717872 : 650
1468440901 2249534301 4654958537 1050792279 6892589235 : 700
e = 2.
7182818284 5904523536 0287471352 6624977572 4709369995 :50
9574966967 6277240766 3035354759 4571382178 5251664274 :100
2746639193 2003059921 8174135966 2904357290 0334295260 :150
5956307381 3232862794 3490763233 8298807531 9525101901 :200
1573834187 9307021540 8914993488 4167509244 7614606680 :250
8226480016 8477411853 7423454424 3710753907 7744992069 :300
5517027618 3860626133 1384583000 7520449338 2656029760 :350
6737113200 7093287091 2744374704 7230696977 2093101416 :400
9283681902 5515108657 4637721112 5238978442 5056953696 :450
7707854499 6996794686 4454905987 9316368892 3009879312 :500
7736178215 4249992295 7635148220 8269895193 6680331825 :550
2886939849 6465105820 9392398294 8879332036 2509443117 :600
3012381970 6841614039 7019837679 3206832823 7646480429 :650
5311802328 7825098194 5581530175 6717361332 0698112509 :700
And if we include phi also in this analysis:
6094370277 0539217176 2931767523 8467481846 7669405132 : 600pi
2886939849 6465105820 9392398294 8879332036 2509443117 :600e
7845878228 9110976250 0302696156 1700250464 3382437764 :600phi
#
0005681271 4526356082 7785771342 7577896091 7363717872 : 650pi
3012381970 6841614039 7019837679 3206832823 7646480429 :650e
8610283831 2683303724 2926752631 1653392473 1671112115 :650phi
So we delete or ignore the decimal point and we ask if the integers of
the
10^603 region of pi, e, and phi would be primes.
Obviously pi is a composite at 10^600 to 10^605. Whereas e maybe a
prime
at 10^602 and 10^604 and phi maybe a prime at 10^604 considering it
has that
"1" digit.
So does anyone know if "e" and phi are primes at 10^604 with that "1"
digit?
Archimedes Plutonium
the first prime in "e" is 271, but
for phi we have to go out to 1618033. So it looks as though phi is
going to have sparse number of primes in
its digit string. But the question on my mind is whether "e" and "phi"
are primes at 604 digit of ending in "1"?
Veky
> Now the first prime in pi is 31 when we ignore the decimal point and
> the first prime in "e" is 271, but
> for phi we have to go out to 1618033. So it looks as though phi is
> going to have sparse number of primes in
> its digit string.
No. Until precision of 10^603, there are exactly 4 primes in decimal initial digits of all three of them.
pi: 3, 31, 314159, 31415926535897932384626433832795028841
e: 2, 271, 2718281, 271828182845904523536028747135266249775724709... ...3699959574966967627724076630353547594571
phi: 1618033, 1618033988749,
1618033988749894848204586834365638117720309179805762862135448622705260...
4628189024497072072041893911374847540880753868917521266338622235369317...
9318006076672635443338908659593958290563832266131992829026788067520876...
689250171169620703222104321626954862629631361,
1618033988749894848204586834365638117720309179805762862135448622705260...
4628189024497072072041893911374847540880753868917521266338622235369317...
9318006076672635443338908659593958290563832266131992829026788067520876...
6892501711696207032221043216269548626296313614438149758701220340805887
> But the question on my mind is whether "e" and "phi"
> are primes at 604 digit of ending in "1"?
No, they are not. floor(10^603*e) is divisible by 251549, and floor(10^603*phi) is divisible by 34327.
You're welcome. :-)
David R Tribble
> Now the first prime in pi is 31 when we ignore the decimal point and
> the first prime in "e" is 271, but
> for phi we have to go out to 1618033. So it looks as though phi is
> going to have sparse number of primes in
> its digit string.
Why not other transcendental numbers, such as sqrt(2) and 1/pi ?
Archimedes Plutonium
literature
to mathematics, but for a compelling motivation. I want to find out
the relationship
of the border of infinity at 10^603 since pi has three zero digits in
a row, and since
pi is related to the primes, how the primes have that three zero
digits in a row, how
they reflect that characteristic. Put in another way, if I were
looking for the border
of finite to infinite by looking only at primes, how would I recognize
10^603 from the
primes as being the infinity border? You see, three zero digits in a
row in pi is a give-away
tell tale sign. But how would the primes communicate that 10^603 is
the infinity border?
So that is the motivation of this new category of primes definition. I
invent three new classes of primes:
pi-fibonacci primes
e-fibonacci primes
phi-fibonacci primes
Fibonacci primes are these:
3, 4, 5, 7, 11, 13, 17, 23, 29, 43, 47, 83, 131, 137, 359, 431, 433,
449, 509, 569, 571, 2971, 4723, 5387, 9311, 9677, 14431, 25561, 30757,
35999, and 81839.
The first pi prime is 31 but there is no 31 as a fibonacci prime. The
next pi prime is
314159. Now I have not yet looked up to see if that is a fibonacci
prime.
The first e-prime is 2718281. Now I need to look up and see if that is
a Fibonacci prime
also.
The first phi prime is 1618033 and I have to see if it is a Fibonacci
prime.
So a pi-fibonacci or e-fibonacci or phi-fibonacci primes are primes
that are simultaneously
primes in both. Now this maybe an empty set for all three, until I can
locate at least one
member.
Phi = 1.
6180339887 4989484820 4586834365 6381177203 0917980576 :50
2862135448 6227052604 6281890244 9707207204 1893911374 :100
8475408807 5386891752 1266338622 2353693179 3180060766 :150
7263544333 8908659593 9582905638 3226613199 2829026788 :200
0675208766 8925017116 9620703222 1043216269 5486262963 :250
1361443814 9758701220 3408058879 5445474924 6185695364 :300
8644492410 4432077134 4947049565 8467885098 7433944221 :350
2544877066 4780915884 6074998871 2400765217 0575179788 :400
3416625624 9407589069 7040002812 1042762177 1117778053 :450
1531714101 1704666599 1466979873 1761356006 7087480710 :500
1317952368 9427521948 4353056783 0022878569 9782977834 :550
7845878228 9110976250 0302696156 1700250464 3382437764 :600
8610283831 2683303724 2926752631 1653392473 1671112115 :650
8818638513 3162038400 5222165791 2866752946 5490681131 :700
7159934323 5973494985 0904094762 1322298101 7261070596 :750
1164562990 9816290555 2085247903 5240602017 2799747175 :800
3427775927 7862561943 2082750513 1218156285 5122248093 :850
9471234145 1702237358 0577278616 0086883829 5230459264 :900
78780178899219902707769038953219 681
So let me repeat the motivation again. I am looking for how to tell
that 10^603 is
the border to infinity from purely the nature of primes. From purely
the nature of
pi, it is easy to see the border as three zero digits in a row, but
now how does
primes tell us it is the border?
Archimedes Plutonium
http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html
Fn130021 2706998033...75321 has 27173 digits
Now that prime is looking awfully close to a e-fibonacci prime if such
a one exists.
But I have the strange feeling that a pi-fibonacci or e-fibonacci or
phi-fibonacci
prime is extremely rare and possibly nonexistant. But then again, most
of the primes
in the 10^603 region have never been revealed.
I did run across an interesting curio from this website:
http://primes.utm.edu/curios/page.php/603.html
The first 603 digits of 603^603 form a 2002-bit prime. [Kulsha]
Archimedes Plutonium
http://primes.utm.edu/curios/page.php/603.html
The first 603 digits of 603^603 form a 2002-bit prime. [Kulsha]
Maybe I do not have to look any further for the primes characteristic
that matches the pi three zero digits in a row at 10^-603. Maybe the
above of 603^603 as the first 603 digits being a prime is all the
answer
of how primes mirror image pi. But if it is, it will require some
patient
interpretation to fathom the connection. It is sort of like a fractal
of
primes, exponents, digit arrangement and digit count. Why would pi at
10^-603 with its three zero digits in a row be connected to the fact
that
603^603 has its first 603 digits being prime? If there is a connection
there,
it escapes me at the moment.
Archimedes Plutonium
<plutonium.archime...@gmail.com> wrote:
> http://primes.utm.edu/curios/page.php/603.html
> The first 603 digits of 603^603 form a 2002-bit prime. [Kulsha]
>
> Maybe I do not have to look any further for the primes characteristic
> that matches the pi three zero digits in a row at 10^-603. Maybe the
> above of 603^603 as the first 603 digits being a prime is all the
> answer
> of how primes mirror image pi. But if it is, it will require some
> patient
> interpretation to fathom the connection. It is sort of like a fractal
> of
> primes, exponents, digit arrangement and digit count. Why would pi at
> 10^-603 with its three zero digits in a row be connected to the fact
> that
> 603^603 has its first 603 digits being prime? If there is a connection
> there,
> it escapes me at the moment.
>
Really incredible that 603^603 existed in the literature before. I
have to check to
see who discovered this curious item? Kulsha? And whether the
discoverer was motivated
by the fact that pi had three zero digits in a row at 10^-603?
But I believe I have a preliminary interpretation of that amazing fact
that 603^603 has
its first 603 digits a prime.
As I repeatedly mentioned, I am looking for the analog of three zero
digits in a row of pi
at 10^-603 to that of primes, since primes and pi are intimately
related. So what does the primes
have that mirror reflects three zero digits in a row to form the
border of finite and infinity?
I believe it is the idea that primes are scarce and rare for the first
time at 10^603 and that a huge
space is prime free. I believe that is the analog of three zero digits
in a row. Think of the zero digit
itself as a prime free zone and think of primes as the building blocks
of future numbers. So that once
you have alot of primes existing below 10^603, that at 10^603 is a
pocket free zone of no primes. Now I do not
mean to imply that the Prime Number theorem is violatated or any of
the other theorems concerning the
prime density or distribution. I simply mean that three zero digits in
a row in pi is no violation of anything
and that primes are rare in this pocket zone of 10^603 because we had
an abundance below 10^603.
So that the number 603^603 has its first 603 digits as prime is a
symptom of the idea that most of these
numbers around 10^603 region are composites.
Now I know that pi has four zero digits in a row further on, so can we
expect a similar number further on call it
z such that z^z has its first z digits as a prime and another pocket
zone free of primes?
So what I was looking for as the analog of pi three zero digits in a
row with something that was prime special, is
not really prime special other than be a prime free zone and that is
what marks infinity. So that the Desargues
parabola meeting a the "point of infinity" is not a prime number but
rather a composite where there is a heavy
concentration of composites at the point of infinity. At least this is
what a preliminary interpretation would make out.
Archimedes Plutonium
The first pi prime is 31 but there is no 31 as a fibonacci prime.
The
next pi prime is
314159. Now I have not yet looked up to see if
that is a fibonacci
prime.
The first e-prime is 2718281. Now I need to look up and see if that
is
a Fibonacci prime
also.
The first phi prime is 1618033 and I have to see if it is a Fibonacci
prime.
So a pi-fibonacci or e-fibonacci or phi-fibonacci primes are primes
that are simultaneously primes in both. Now this maybe an empty set
for all three, until I can
locate at least one
member.
Now in Old-Math preceded over by those that are too daffy and dull to
precision define finite versus infinite, upon seeing the first two pi-
primes of 31 and 314159 would instantly make their Old Math mistake of
asking whether
pi primes is a finite set or infinite set? This question when then be
placed into the record books as being a open conjecture, along with
thousands of other silly unproven conjectures. The word "silly" is
apt, because these are not
legimate math questions, because finite versus infinite was never
given a precision definition.
In New-Math, the instant pi-primes or pi-fibonacci primes or e-primes
or e-fibonacci primes is defined and seeing that
it is impossible to gather together 10^603 specimens of these primes
in the interval 0 to 10^1206, the instant that is
recognized, we immediately know that these prime sets are finite. A
different picture for the regular primes or twin primes or quad primes
or hex primes emerges, becuase those primes have sets of x/Ln(x) or x/
zLn(x) as their density of distribution and thus a huge set of more
than 10^603 of those types of primes can be gathered from 0 to
10^1206.
So infinity is really density, and that is what emerges when a
precision definition is given to infinity. Infinity is precision
defined as the border between finite and infinity, and once that
border is staked out as 10^603, we have another border of algebraic-
completeness 10^1206.
So in Old Math, they need warehouses to store all their conjectures of
whether set A or B are finite or infinite. In New Math, as fast as you
can define new sets of primes, I can tell you if they are finite or
infinite almost as fast as you blurt them out.
And that is the way mathematics should really be. If math has its
house in order, since it is the science of precision,
it should conquer every problem presented in a short period of time.
If math has little order and precision, it has problems centuries and
milleniums old. Shame that mathematics cannot sign up for Social
Security for it would be richer than the government itself due to its
inability to solve what it should solve.
