The real 10-adic integers
Patrick Reany
wanted to find a way to prove or disprove the conjecture based on noting
the effect of exponentiation on the rightmost digits of ordinary positive
integers. What he discovered (or possibly rediscovered) is some theorems
of the 10-adic integers. DeLugt got me interested in the project to
writeup his results and publish them.
Now, DeLugt and I did not see eye-to-eye on what the 10-adics really
were. I wanted to treat them as purely formal and treat them as an
unordered, and without magnitude, set of formal entities. He disagreed and
I found out how easy it is to generate controvery over 10-adic integers.
The one thing we could agree on is that the four idempotents that he
discovered in the set are fascinating, and we couldn't understand how the
establishment in math couldn't find them interesting. Oh well.
Consider the set of all strings of digits (0..9) formed by taking a first
digit (the rightmost) and a second and a third, and so on without end.
Each of these strings is called a transfinite digital (td) number, and
each is an ordinal number of type omega, to use Cantor's notation. We can
define addition and multiplication on the set of these td's in the obvious
way. Now we can add into the set the additive inverses of each td, and
then we get a commutative ring, which I called the ring of Hyperintegers.
Now, inspite of the fact that the ordinary integers are isomorphic to a
proper subring of the hyperintegers, I do not consider this too important,
and I do not attempt to form an ordering, magnitude, or valuation on the
set of hyperintegers.
An obvious question is whether or not the operations of addition and
multiplication we've defined on the hyperintergers are well defined. But
they are, because to be well defined all we need do is to be able to
establish the ith digit in the hyperinteger for any positive natural i.
Let p_i(a) be a function which gives us the ith digit of the hyperinteger
a. Let r_i(a) be the i rightmost digits of a, maintaining their order.
THEOREM: There are four idempotents in the hyperintegers.
Definition: An _idempotent_ is an number whose square is equal to itself.
It's easy to show that ...0000 and ...0001 are idempotents of the
hyperintegers; our task is to produce the other two. (Unless stated
otherwise, assume that three or more digit patterns repeated side-by-side
mean that they are repeated infinitely.)
Just as in other number systems, the hyperintegers has its share of
multiple numerals that represent a given hyperinteger. For instance, we
can write ...000 as 0, and ...0001 as 1.
Definition: A hyperinteger numeral is said to be in _proper_ form if each
of its entries is a digit (0..9). (Of course, 0 and 1 are improper forms
for hyperintegers.)
Definition: Two td's a,b are said to be equal iff when both a,b are given
as proper numerals, for all i in Naturals,
p_i(a) = p_i(b).
From this last definition we can prove that
...000 = ...999X
where X is the equivalent to '10', only not precisely. Think of X as an
instruction to replace itself by '0' and carry a 1 to the next left digit
place.
Now we can define the complement function, comp(a), for all hyperintegers.
For a a positive hyperinteger comp(a) := ...9999X - a, and for a a negative
hyperinteger, comp(a) := -(...999X + a). Obviously, for all td's
x + comp(x) = 0
thus the set of td's also forms a rings without formally introducing the
negative sign.
We need to list some obvious properties of the r_n() function to help us
out. Think of the r_n() function as a mapping of hyperintegers into the
integers mod 10^n. Then:
r_n(x) = sign(x) p_n(x) p_{n-1}(x) . . p_1(x)
where the above represent digits, not multiplication.
r_n(r_n(x)) = r_n(x)
r_n(x+y) = r_n(r_n(x)+r_n(y)) = r_n(x)+r_n(y) (addition taken mod 10^n)
r_n(x.y) = r_n(r_n(x).r_n(y)) = r_n(x).r_n(y) (multiplication mod 10^n)
Although some liberties were taken above, it should be easy to keep from
getting confused by context.
Now, if x is a positive hyperinteger, then x is idempotent if for all
Naturals i, either
p_i(x^2) = p_i(x),
or
r_i(x^2) = r_i(x).
when both x and x^2 are given in proper form.
It is not difficult to show that idempotents starting with 0 or 1 are
unique, and it is easy to show that no idempotents can start with
{2,3,4,7,8,9}. This leaves the possibility of idempotents starting with 5
and or 6.
THEOREM: There exists a positive hyperinteger idempotent whose first
(rightmost) digit is 5.
Proof:
First we generate the hyperinteger then show that it satifies the
definition of idemotency.
Let e_1 be given recursively by
p_1(e_1) = 5
p_{n+1}(e_1) = p_{n+1}([r_n(e_1)]^2)
Now we proceed by induction:
e_1 obviously satifies the requirement for i = 1.
Now assume that r_k(e_1^2) = r_k(e_1) for arbitrary k (the inductive
hypothesis), and then show that the equation holds for k -> k+1 as well.
Now, r_{k+1}(e_1^2) = r_{k+1}([r_{k+1}(e_1)]^2).
Let r_{k+1}(e_1) = d.10^k+r_k(e_1) where d = p_{k+1}(r_{k+1}(e_1))
then
r_{k+1}(e_1^2) = r_{k+1}(d^2 10^{2k}+2d10^k r_k(e_1)+[r_k(e_1)]^2)
= p_{k+1}([r_k(e_1)]^2)10^k + r_k([r_k(e_1)]^2)
= p_{k+1}([r_k(e_1)]^2)10^k + r_k(r_k(e_1)) by inductive hyp
= p_{k+1}(e_1)10^k + r_k(e_1) by defn of e_1
= r_{k+1}(e_1)
The first step is justified by the fact that for any Natural number k,
the first two terms are in the kernel of the mapping r_{k+1}(). The first
term is obviously so, but the second is too because the factor of 10^k
gets a multiple of at least another 10 because of the product 2.r_k(e_1),
since p_1(e_1)=5. This leaves a factor of 10^{k+1} which makes the second
term in the kernel of r_{k+1}().
In 1985 or 1986 I calculated the first 100 digits of e_1. I present the
first 50: ...5742342323 0896109004 1066199773 9225625991 8212890625.
In a ring with identity, idempotents come in pairs: Define e_2 := 1 - e_1.
Then e_2^2 = (1 - e_1)^2 = 1 - 2e_1 + e_1 = 1 - e_1 = e_2. We can get e_2
in positive form by using that
e_2 = 1 + comp(e_1)
= 1 + ...375
= ...376
The first 50 digits of e_2 are:
...4257657676 9103890995 8933800226 0774374008 1787109376
It is trivial that
e_1^n + e_2^n = 1^n for all Natural numbers n.
But this is not an invalidation of number theory based on ordinary
integers, and it certainly is not an invalidation of FLT, which is a
theorem based on ordinary integers.
But do the hyperintegers have any practical uses? I think so. One is for
introducing the concept of rings to math student, particularly high
school students---even at the risk of them going overboard on them as
some have done. Another use is in the calculation of the rightmost digits of
integers taken to large powers. Consider the problem of finding the n
rightmost digits of 8573647^385645 (digits chosen at random).
r_n(8573647^385645) = r_n((8573647e_1)^385645) + r_n((8573647e_2)^385645)
the advantage of which is that by "projecting" the base number onto the
idempotents allows us to take the exponent mod the order of the cyclic
multiplicative group formed by multiplying all integers from 0 to 10^n-1
times each idempotent, giving a tractable cyclic group for both e_1 and
e_2, as long as n is not too large.
Finally, of what importance is it that the hyperintegers contain non
trivial idempotents, other than already discussed? Because these
non-trivial idempotents are idempotent pairs, their product is zero, even
though neither of them is zero: e_1e_2 = e_1(1-e_1) = e_1 - e_1 = 0. Thus
the ring of hyperintegers is not a field. So what! Fields aren't the only
objects worth study. Lots of rings are interesting and useful, especially
rings with non-trivial zero-divisors, like the hyperintegers!
cheers
-- Patrick Reany (re...@xroads.com)
Douglas J. Zare
>[...]
>Finally, of what importance is it that the hyperintegers contain non
>trivial idempotents, other than already discussed? Because these
>non-trivial idempotents are idempotent pairs, their product is zero, even
>though neither of them is zero: e_1e_2 = e_1(1-e_1) = e_1 - e_1 = 0. Thus
In fact, e1+e2=1: (x-e1)(x-e2) = x^2-x.
>the ring of hyperintegers is not a field. So what! Fields aren't the only
>objects worth study. Lots of rings are interesting and useful, especially
>rings with non-trivial zero-divisors, like the hyperintegers!
I agree that algebras might be worth more study in some circumstances. In
this case, however, it would clarify the results you mention to note that
the 10-adics are the direct sum of the 2-adics and the 5-adics.
As the 2-adics and 5-adics form fields, the only idempotents are the
obvious ones. To get the idempotents in the 10-adics, one can use the
Chinese Remainder Theorem.
0 is 0 mod 5^n for all n, and 0 mod 2^n for all n.
e1 is 0 mod 5^n for all n, and 1 mod 2^n for all n.
e2 1 0
1 1 1
This (and more) was written up in an accessible article which appeared in
Quantum a couple of years ago.
BTW, I recommend against trying to order them. Also, to make the p-adics a
field, you have to allow finitely many digits to the right of the decimal
point, i.e., p's in the denomenator. Otherwise, you have one of the
canonical examples of a local ring (ring with unique maximal ideal), but
not a field.
Douglas Zare
Terry Tao
Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
>
> There will be many who will come out of the woodwork to try to steal
>the great new idea which started the new millenium in math. The idea
>that Naturals = Adics = Infinite Integers.
Ah yes, but are Counting numbers = Adics? Show us a set with ...121212
elements, Archie baby!
Terry
Terry Tao
Patrick Reany <re...@xroads.com> wrote:
>Now, DeLugt and I did not see eye-to-eye on what the 10-adics really
>were. I wanted to treat them as purely formal and treat them as an
>unordered, and without magnitude, set of formal entities. He disagreed and
>I found out how easy it is to generate controvery over 10-adic integers.
>The one thing we could agree on is that the four idempotents that he
>discovered in the set are fascinating, and we couldn't understand how the
>establishment in math couldn't find them interesting. Oh well.
Well, the main reason is that these idempotents are trivial in the more
useful representation of the 10-adics, which is as the direct sum of the
2-adics and the 5-adics. The idempotents are then (0,0), (0,1), (1,0),
and (1,1).
More generally, all the composite -adics are isomorphic to the direct
sum of various p-adics. As such, only the p-adics are really
interesting to study; any other result involving other -adics can be
easily reduced to the p-adic theory.
Your formalist approach is the correct one - or at least, it is the
correct place to start. The 10-adics are uniquely constructed as formal
digit strings; there is no a priori justification for assuming extra
properties about them. One would have to either prove these properties
from the digit string foundation, or construct (or intuitively describe)
an alternative foundation where these properties were manifestly true
(although one might then have to prove that this new construction was
isomorphic to the old one).
Usually Z_10 is defined to be the inverse limit of the rings Z/(10^n Z)
with the natural projection maps. This gives it a natural ring
structure and a topology, but everything else one has to work for.
>
>Consider the set of all strings of digits (0..9) formed by taking a first
>digit (the rightmost) and a second and a third, and so on without end.
>Each of these strings is called a transfinite digital (td) number, and
>each is an ordinal number of type omega, to use Cantor's notation.
Actually, this last statement is false; there is no constructive way to
order the p-adics such that every p-adic has countably many
predecessors.
A naive extension of the ordering of the natural numbers will run into
difficulties. Which is larger, ....12121212 or ....21212121? There is
no way to totally order the p-adics in such a way that multiplication and
addition preserve order (as they do in the naturals). One possible
ordering is reverse lexicographic (so that the last digit is the
most significant), although this does not reflect the usual ordering of
N (but it does reflect the representation of the p-adics as a
Cantor set), and has no useful properties with respect to addition and
multiplication.
Terry
Santiago Arteaga
Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
>re...@xroads.com (Patrick Reany) writes:
> So, please, by all means come out of the woodwork, but forget your
>motivation of trying to hog and steal priorities.
You may have not read this sentence:
>>Now, inspite of the fact that the ordinary integers are isomorphic to a
>>proper subring of the hyperintegers, I do not consider this too important,
>>and I do not attempt to form an ordering, magnitude, or valuation on the
>>set of hyperintegers.
It is clear that Patrick believes that the naturals and the
hyperintegers are different sets. But I can also see how this sentence is not
so explicit as to avoid your paranoid fear of being stolen your wonderful
discoveries. So I think you might find this story interesting.
When I was a kid, one day I realized that 25^2 ends in 25. So I had
a question: is there a number with three figures whose square ends in itself?
And by trial and error, playing with a calculator, I found out that the
answer was 625. And you may not believe it, but next I wondered whether there
was a number with 4 or 5 or 6 figures whose square ends in itself; I don't
remember how far did I get with a hand calculator, but I was able to guess
that maybe you could find such a number with infinitely many figures.
Now, being a kid, I couldn't find it, I didn't even think of proving
that it existed, and I could not imagine that such a number was unique, or
that there was a similar number ending in 6. Nor could I think of defining
operations with these numbers, nor did I know of Fermat's theorem, nor did
I bother to write anything down.
The point of the whole story is that anybody can think of integers
with infinitely many figures. It is just not so hard. I guess that many
babylonians did it in a boring rainy day thousands of years ago.
So it is just plain pathetic that you accuse somebody of plagiarism
and hogging just because they say that they have played with integer
numbers with infinitely many digits. When somebody shows up claiming that
the integers are equal to the p-adics, then you both can fight forever over
the priority of such a transcendental result, but meanwhile you might
concede us the ability of having curiosity about infinite numbers.
And don't bother condemning me again to your hell, I'm already
there and it's a cozy place.
Santi
Archimedes Plutonium
re...@xroads.com (Patrick Reany) writes:
> In 1985, Anthony DeLugt--an associate of mine--was working on FLT. He
> wanted to find a way to prove or disprove the conjecture based on noting
> the effect of exponentiation on the rightmost digits of ordinary positive
> integers. What he discovered (or possibly rediscovered) is some theorems
> of the 10-adic integers. DeLugt got me interested in the project to
> writeup his results and publish them.
There will be many who will come out of the woodwork to try to steal
the great new idea which started the new millenium in math. The idea
that Naturals = Adics = Infinite Integers.
To Mr. DeLugt and Mr. Reany, it was known a long time before you two
that counterexamples to FLT existed in adics.
But noone before me had the brains to make the final leap, the final
beautiful jump. To say that Naturals = Finite Integers was hogwash. And
to say that Naturals = Adics. Noone in the published literature ever
said Naturals = Adics = Infinite Integers and PROVIDED a published
proof thereof. I was the first to do this starting 1993 here on
Internet.
Archimedes Plutonium
t...@rugola.princeton.edu (Terry Tao) writes:
> Ah yes, but are Counting numbers = Adics? Show us a set with ...121212
> elements, Archie baby!
>
> Terry
Say Terry, Monsieur Tao, do I pronounce your last name Tao like in
Towel or Dao like in dowel stick? You must have met Geison down
there carrying his lastest "Pasteur the Thief and Liar" And I hope
you Terry have kept a "notebook" on Wiles for future biographic
information for Geison. And if not for anything I can use it for
restaurant stand-up comic humor. People in restaurants in California
while eating pastries or club sandwiches will laugh at the dry humor
and jokes of Wiles panache. That same humor flew over the heads of
math people in Cambridge in 1993, but the camera got Wiles laughing as
seen in NYT. I don't know about Wiles as a comedy hit Terry, I seen
to laugh more at the lost time to graduate students there at Princeton,
especially Brian Conrad. His graduate years can be summed up as "study
under a stupor of Wiles". Look out for the pie in your face Terry. But
I have one nice thing to say about the whole Princeton black hole of
1993-1995. Will Schneeburger down there is very smart and I nominate
him to take over after the deadwood the Faltings, Wiles, Kochen and
Conway retire out to pasteur.
Mike Davis
Could somebody recommend a good book that defines -adics and discusses
them in an elementary way?
Thanks.
-mad
--
signoff
Mike Davis == mda...@pobox.wellfleet.com == +1 508 436 8016
Douglas J. Zare
>t...@rugola.princeton.edu (Terry Tao) writes:
>> Ah yes, but are Counting numbers = Adics? Show us a set with ...121212
>> elements, Archie baby!
>> Terry
>
> Say Terry, Monsieur Tao, do I pronounce your last name Tao like in
>Towel or Dao like in dowel stick? You must have met Geison down
>him to take over after the deadwood the Faltings, Wiles, Kochen and
>Conway retire out to pasteur.
Has anyone else been collecting the questions AP can't answer and in
response to which he insults people?
Douglas Zare
Lee Rudolph
>Has anyone else been collecting the questions AP can't answer
...or (all we can be *sure* of) leaves unanswered:
the sci.math.plutonium FUQ.
I see a great need.
Lee Rudolph
Archimedes Plutonium
>> Ah yes, but are Counting numbers = Adics? Show us a set with ...121212
In article <3qlcn5$d...@gap.cco.caltech.edu>
za...@cco.caltech.edu (Douglas J. Zare) writes:
> Has anyone else been collecting the questions AP can't answer and in
> response to which he insults people?
Oh Dougy Baby, you have to get used to the Net faster than that.
Terry asks me that question periodically. I answered it a long time ago
by asking him to define "counting" since counting is a foggy notion as
per adics. And then give or take some months Terry asks the same
question, perhaps changing a few words or digits, and pretending as if
I had never answered it or my answer was no good. It is a nice
propaganda tactic. And I think I will use it in my bag of tricks. And I
feel it would drive you bonkers if Terry or I played the same trick on
you. To give you an example Douglas. I could do it with your p-adic
answer. BTW, your answers sound like John Baez when I read some of his
posts a long time ago but no more. Answers that are never direct, never
full, just beating around the bush. I do not like it when I ask someone
for a need to know question and they give indirect slop, as if I want
to play cat and mouse. In fact I think I will save your post right here
and now so I have it handy.
In article <3qlc0p$d...@gap.cco.caltech.edu>
za...@cco.caltech.edu (Douglas J. Zare) writes:
> Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
> >Danny Calegari <dan...@mundoe.maths.mu.oz.au> writes:
> >>[...]
> >> above. I wonder whether such strings do exist for all p, or for what p
> >> they exist. I think this is at least an entertaining question, even if
> >> it does not shed much light on FLT.
> >[...]
> >From what I gathered, I recall that
> >idempotents require n-adics and whether they require n-adics which are
> >divisible by two Real primes (Real primes such as 2.00 and 5.00 for
> >Real 10.000) and not say three Real primes like in 70-adics.
> >[...]
>
> Note that it suffices to consider square-free n, e.g., 4-adics = 2-adics.
> If n=(p1)(p2)...(pk), then n-adics are the direct sum of pi-adics for
> 1<=i<=k.
>
> p-adic integers form an integeral domain for p prime and x^2-x factors
> as x(x-1). Thus, the only idempotents in the p-adics are 0 and 1.
>
> By the Chinese Remainder Theorem, there are 2^k idempotents in the
> n-adics. Two of these are 0 and 1.
>
> Douglas Zare
There it be. Now Douglas in say 2 or 3 months when I am with free
time you may perhaps see a post like this whereever you make some other
post refering to p-adics.
In article <3qlc0p$d...@gap.cco.caltech.edu>
za...@cco.caltech.edu (Douglas J. Zare) writes:
[lines deleted]
> n-adics are the direct sum of pi-adics
And of course I will ask you Dougy Baby in that future post whether
it is e-adics and not pi-adics? Do you see the cute trick Terry likes
so much to play. And when you catch on to this nifty trick it is good
for a few laughs but with your thick skin Dougy, I think this trick
played on you would drive you bonkers fast.
But seriously now Douglas, I want to publish this "The Worlds First
Valid Proof Of Euclid's Indirect Proof Of The Infinitude Of Primes" for
I feel that virtually all the Euclid proofs stated in books, even
Hardy's A MATHEMATICIANS APOLOGY are wrong and invalid for they argue
that there is a prime factor that may be missing. I argue in my proof
that (P1 x P2 x ... x Pk) + 1 is necessarily prime. So Douglas, the
joking aside, is there a place at Caltech to mail this gem of mine, or
do you recommend someone else who will review it? Since I presume you
must have been published yourself? The reason I have been putting off
this submission is obvious, since I have proved Naturals = Adics =
Infinite Integers it is quite embarrassing for me to submit to any
journal since there are no primes at all in adics. And that prime
numbers was just a foggy notion. So I can't run around saying I have
given the first valid indirect Euclid proof when no primes exist
whatsoever. Can I?
And lately I have considered as to the reason why there is such a
logical gap between my valid proof of Euclid's Infinitude of Primes and
fake proofs such as Hardy's or Montgomery&Niven&Zuckerman. What I am
thinking is that between the discrepancies of looking for factors and
my method of MP's+1 as necessarily prime, that this gap may indicate
that all proofs of infinitude of primes are bogus. That all proofs of
IP have fatal flaws since no primes exist at all in adics.
So Douglas does Caltech have a math journal? If not why not? And who
do you recommend that I send my IP corrected of the Hardy fakery?
Douglas J. Zare
>[...]
>Answers that are never direct, never
>full, just beating around the bush. I do not like it when I ask someone
>for a need to know question and they give indirect slop, as if I want
>to play cat and mouse.
I had written the following:
>> If n=(p1)(p2)...(pk), then n-adics are the direct sum of pi-adics for
>> 1<=i<=k.
>>[...]
>> By the Chinese Remainder Theorem, there are 2^k idempotents in the
>> n-adics. Two of these are 0 and 1.
>>[...]
How was this not a direct answer? The Chinese Remainder Theorem is
constructive. See any introductory book on number theory or algebra.
Try this question to see if you understand my post now:
What are the possible factorizations of x^2-x over the n-adics?
> There it be. Now Douglas in say 2 or 3 months when I am with free
>time you may perhaps see a post like this whereever you make some other
>post refering to p-adics.
>
>za...@cco.caltech.edu (Douglas J. Zare) writes:
>[lines deleted]
>> n-adics are the direct sum of pi-adics
>
> And of course I will ask you Dougy Baby in that future post whether
>it is e-adics and not pi-adics?
This is out of context. I said, "n = (p1)(p2)...(pk)"..."n-adics are the
direct sum of pi-adics for 1<=i<=k." Once again, _i_was_the_index_. If you
have trouble understanding English, you might want to get someone help
you to translate things like this.
>[...]
> But seriously now Douglas, I want to publish this "The Worlds First
>Valid Proof Of Euclid's Indirect Proof Of The Infinitude Of Primes" for
>I feel that virtually all the Euclid proofs stated in books, even
>Hardy's A MATHEMATICIANS APOLOGY are wrong and invalid for they argue
>that there is a prime factor that may be missing. I argue in my proof
>that (P1 x P2 x ... x Pk) + 1 is necessarily prime.
Did you not understand/believe the counterexamples?
>So Douglas, the
>joking aside, is there a place at Caltech to mail this gem of mine, or
>do you recommend someone else who will review it? [...]
There are professors here who often receive material like that. I
suggest sending it to Caltech/175-37/Pasadena CA 91125 if you must.
Since mathematics is so fuzzy and unclear to you, I recommend taking some
courses at a community college, applying to a 4-year college, and then
completing a masters program before seriously attempting fundamental
research. It is rare for anyone below the undergraduate level to do
worthwhile or even publishable research.
Douglas Zare
Archimedes Plutonium
lrud...@panix.com (Lee Rudolph) writes:
> za...@cco.caltech.edu (Douglas J. Zare) writes:
>
> >Has anyone else been collecting the questions AP can't answer
>
Archimedes Plutonium
mdavis@eddie (Mike Davis) writes:
>
> Could somebody recommend a good book that defines -adics and discusses
> them in an elementary way?
There is none. I have sent out the alarm clock on making two books.
Two Schaums Outline type of books, not the heavily dosed math
terminology, one written on just 10-adics alone where a high school
student can practice adding, mult, subt, and divid. Another outline
book on adics in more general.
No elementary book exists. Why does none exist? Because before 1993
adics were considered just unimportant extensions of Natural numbers.
But in 1993 I discovered that the Naturals are the adics. The old time
definition of Naturals as finite integers was not defined precisely and
can never be defined. Naturals = Adics = Infinite Integers. This
discovery is so new that the math community has not had the time to
fill the gap of elementary books on adics. But they will come.
Archimedes Plutonium
za...@cco.caltech.edu (Douglas J. Zare) writes:
> How was this not a direct answer? The Chinese Remainder Theorem is
> constructive. See any introductory book on number theory or algebra.
> Try this question to see if you understand my post now:
>
> What are the possible factorizations of x^2-x over the n-adics?
I am talking about the conversion table I asked for last month from
Eucl geom to Loba geom and vice versa. And you never posted it; you
only posted irrelevant quiz questions saying that if I answered those
offbeat questions you might out of some change of heart answer me
directly. Many days transpired and then my good friend gave me the
information and I posted it to sci.math. What do you do after I post
that conversion table? Here is your excuse----
In article <3qo3co$6...@gap.cco.caltech.edu>
za...@cco.caltech.edu (Douglas J. Zare) writes:
> Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
> >[...]
> >there are more people out there who care more about
> >hate than they ever care about doing math.
> >[...]
>
> Recently, you have accused people of rudeness, idiocy, and performing
> publicity stunts. Will you next complain about grammar and inappropriate
> posts?
>
> >[quoting a friend]
> >I don't know if there's a standard coordinate notation for the
> >Lobachevskian plane.)
> >
> >Now you can use the following mappings:
> >To map E -> R: theta{R} = theta{E}, r{R} = tan r{E}
> >To map R -> E: theta{E} = theta{R}, r{E} = arctan r{R}
> >To map E -> L: theta{L} = theta{E}, r{L} = tanh r{E}
> >To map L -> E: theta{E} = theta{L}, r{E} = arctanh r{L}
>
> Check these again. The first two, at least, must be switched.
>
> >These mappings are line-preserving, so a triangle of one plane
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> Check this again. Map x=1 to S^2.
> Which model are you using for H^2? If r is the intrinsic distance, the
> map is not even reasonable.
>
> >will map to a triangle of another plane.
> >---- end of my friend[']s teachings ----
>
> At least these were almost pointwise maps.
>
> >[...]
> >And Zare, aims to waste my time with his
> >joke math questions as if I am here in sci.math taking a College
> >Freshman test.
> >[...]
>
> I did not give you something like the above because you would not
> understand it (among other reasons). In fact, you did not understand your
> friend's attempt enough to see it was flawed.
>
> > Hey Zare, I have some time now for your childish[!] posts. I want you
> >to post your version of Euclid's Infinitude of Primes, indirect proof.
> >Come on Douglas Zare, I want to reveal to the math world, since you are
> >plaguing my threads with your drivel[!]. Come Douglas Zare, post your
> >version of the Euclid Infinitude of Primes, and if you do not, then do
> >not ever again post to any of my threads.
>
> You did start complaining about inappropriate posts! Now if only I have a
> mirror... BTW, "reveal" is a transitive verb. You did not express a complete
> thought. If you still did not understand the many correct proofs that others
> gave, I will see what I can do to put them in a language you can understand.
> Finally, if they are your threads, keep them on your account and do not
> post them.
>
> Douglas Zare
----- end of DZ post -----
And BTW, Zare, today I received an email by you which was routed to
my sysadm. Your email is a complaint on me. So, Zare, are you the one
behind my "fast-forward-recycled-out" post? That is, I wanted to post
to my Doubly Infinite thread which I had posted and 3 days later it had
been removed. Whereas the normal lifespan of a post is 5 days, that
post of mine that I wanted to add the conversion table had been killed.
Are you Zare, the person directly responsible for the shortened
lifespan of that post? That thread? I know I had put a song to that
thread, was it the song that repulsed you? The song "Oh how I love
Jesus" turned into "Oh How I Love PU". Was it that song that gave you
or someone else the authority to kill my post and thread?
BTW, Zare, you are wrong about tan and arctan, just think about
symmetry from E->R and E to L. But then math is not your strong point
anyway, aren't you a philosopher Douglas? Or the side window attendant
at a MacDonald's drive through?
Archimedes Plutonium
> Did you not understand/believe the counterexamples?
No, Hardy's proof in A MATHEMATICIANS APOLOGY, and
Montgomery&Niven&Zuckerman 's proof in Number Theory text are both
wrong and flawed.
Douglas, when you take the finite set of all primes existing and
multiply them and add 1, that new number is NECESSARILY prime.
Counterexamples mean nothing in this pure slab of logic proof. I am the
first to uncover the illogic of so many popular renditions of Euclid's
Infinitude of Primes indirect proof. Hardy and M&N&Z have flawed and
invalid proof of IP.
I have not attempted to publish this as of recently, for I intend to
pull something even greater out of the fire. Since Naturals = Adics =
Infinite Integers there are NO primes existing at all in Naturals. I am
thinking that the discrepancies, the fault or fissure between so many
variants of IP is due to the fact that none of those proofs are valid.
This post in 1993 has bothered me ever since.
---- start of Peter's post -----
From: Peter...@hg.maus.de (Peter Eckel)
Newsgroups: sci.math
Subject: Re: PROOF OF THE INFINITUDE OF
Message-ID: <A11...@HG.maus.de>
Date: Sun, 29 Aug 93 10:19:00 GMT
References: <25ock4$7...@mathj.usc.edu>
Organization: MausNet (Mitglied im IN e.V.)
Lines: 14
B> If W-1 or (mutatis mutandi) W+1 is not prime then it has a prime
B> factor. But none of p1,p2, . . ., pL is a divisor, and these are
the
B> only primes; contradiction; so W-1 and W+1 are prime.
It depends on when we accept that our assumption that p1,. . .,pL are
the only primes is wrong. At the moment we have stated that W-1 and
W+1 (BTW, for what purpose do we need both of them?) are prime our
assumption leads to a contradiction, and we
can't prove anymore that they are prime. This doesn't look very
elegant to me.
On the other hand, when we just say that by the properties of W-1 and
W+1 there has to be a larger prime than pL, we do not have to change
anything.
Regards, Peter.
------ end of Peter's post -----
Since Adics have no primes and since I believe Naturals = Adics =
Infinite Integers. Then something has to be done about the proof of
the infinitude of primes. What Peter's post suggests is that some math
proofs have a time element built into the steps of the proof. Since
there are so many versions of IP such as me saying that Hardy's is
wrong and my NECESSARILY prime (the number mult the lot add 1) version
correct. I had always thought that all math proofs should be able to be
BOILED down into symbolic logic. But symbolic logic cannot even handle
a simple proof such as IP. So it is the TIME element itself in math
proofs that need to focus attention on.
It is my hunch that all proofs of the infinitude of primes are fakery
because no primes exist in adics. And it was this Time factor that
placed an illusion that the proof was true but in fact without the time
element of arranging the actual steps of the proof that the proof of IP
is never doable.
To give you an example of time element in a math proof, the steps
that go to making the proof have to go in some kind of order. If we
cannot place in proper order the steps of IP then IP does not exist.
And Adics have no primes confirms that no Euclid proof of infinitude of
primes exists.
Leong Weng Ng
Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
>
>Douglas, when you take the finite set of all primes existing and
>multiply them and add 1, that new number is NECESSARILY prime.
Please tell me your definition of a prime and if any such primes exist
please give me the first 3 primes. Also, please define a Natural, an Adic
and an Infinite Integer and give one example of each. Also, please tell me
what you would call a number that looks like -4, 4, or 41. Thanks for
your clarification.
[Other stuff deleted for brevity]
W.L.
--
____________________________________________________________________________
Ng Weng Leong E:--------|-------|---------|------|
Mail Add: PO Box 2285 Stanford Ca 94309 B:---7-10-|-8-7---|-----7---|---7--|
Dorm Add: Rm 336 Roble Hall (3rd floor) G:---7----|---6-6-|-----7-7-|---6--|
Phone # : (415) 497-4161 D:---9----|---7---|--9-11---|-7----|
Favorite Greek Symbol: small zeta A:10------|-5-----|-10------|-5----|
Second Favorite Greek Symbol: small xi E:--------|-------|---------|------|
____________________________________________________________________________/
Dik T. Winter
> But seriously now Douglas, I want to publish this "The Worlds First
> Valid Proof Of Euclid's Indirect Proof Of The Infinitude Of Primes" for
> I feel that virtually all the Euclid proofs stated in books, even
> Hardy's A MATHEMATICIANS APOLOGY are wrong and invalid for they argue
> that there is a prime factor that may be missing.
But you never gave a definition of prime! With the current definition
and naturals = adics, there are no primes, so how can you proof an
infinitude of primes? For instance, suppose 2 and 5 are all primes we
now. Now in that case 2*5 = 11 is prime. But 11 is not prime because
it is divisible by 7 (11/7 = ...1428571428573) and 7 is not prime either
because it is divisible by 11 (7/11 = ...6363636363637).
What gives?
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924098
home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: d...@cwi.nl
Archimedes Plutonium
d...@cwi.nl (Dik T. Winter) writes:
> But you never gave a definition of prime! With the current definition
> and naturals = adics, there are no primes, so how can you proof an
> infinitude of primes? For instance, suppose 2 and 5 are all primes we
> now. Now in that case 2*5 = 11 is prime. But 11 is not prime because
> it is divisible by 7 (11/7 = ...1428571428573) and 7 is not prime either
> because it is divisible by 11 (7/11 = ...6363636363637).
>
> What gives?
Well, please give me some time for I feel there is some rich gold
deposits between the Hardy proof of either (P1 x P2 x . . . x Pk) + 1
is prime itself or it has a prime factor not in the list of primes of
all supposed primes {P1,P2, . .,Pk} which I feel is a flawed proof. And
my proof attempt of IP which says that (P1 x P2 x . . . x Pk) + 1 is
NECESSARILY prime. So forget about looking at a prime factor as does
Hardy or Montgomery&Niven&Zuckerman do.
Dik, I know what you are saying that the adics have no primes and
that with Naturals = Adics there should be no infinitude of primes. And
I strongly believe that is correct, that no primes at all exist. So,
what I am thinking is that since there are so many versions of IP
proofs, perhaps between the gaps of Hardy's proof attempt of Infinitude
of Primes and all others there are these variances because they are
discrepancies. There should not be these many variances to one simple
proof and so this might be because there is no proof in the first
place. I am just exploring that possibility.
Archimedes Plutonium
> >Hardy's A MATHEMATICIANS APOLOGY are wrong and invalid for they argue
> >that there is a prime factor that may be missing. I argue in my proof
> >that (P1 x P2 x ... x Pk) + 1 is necessarily prime.
>
> Did you not understand/believe the counterexamples?
What counterexamples? That number (P1 x P2 x ... x Pk) + 1 is
necessarily prime in the indirect proof.
Archimedes Plutonium
alg...@leland.Stanford.EDU (Leong Weng Ng) writes:
> In article <3qq8tf$1...@dartvax.dartmouth.edu>,
> Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
> >
> >Douglas, when you take the finite set of all primes existing and
> >multiply them and add 1, that new number is NECESSARILY prime.
>
> Please tell me your definition of a prime and if any such primes exist
> please give me the first 3 primes. Also, please define a Natural, an Adic
> and an Infinite Integer and give one example of each. Also, please tell me
> what you would call a number that looks like -4, 4, or 41. Thanks for
> your clarification.
>
> [Other stuff deleted for brevity]
>
> W.L.
Mr. Ng, I am not throwing any curves here, so all the definitions you
ask for are the definitions which are currently accepted. For the
primes it is the set {2,3,5,7,....}.
I had posted this deal about Euclid's infinitude of primes (IP) proof
several times to sci.math 1993-1994.
The reason I pull it up again just now is because with my assertion
that Naturals = Adics = Infinite Integers, there are no primes as Dik
so well shows. So, this is the reason I rehash it here for I think
that perhaps the gulf between the many variants on the proof of IP is
because there is no valid proof of IP. That all of those proofs are
flawed. And the reason they are flawed is because no primes exist. In
other words we will have to go back into the definition of primeness
and make it more precise. This makes sense in another vain. Perfect
numbers are vague because noone has really precisely hammered out the
factor set of a number and that perhaps is why infinitude of perfect
numbers and odd perfect number proofs are a hopeless quagmire.
Hardy's proof went like this. Mult the lot add 1, either this number
is prime or a prime factor not in original set exists, contradiction,
proof. My proof, which says that Hardy's IP is flawed goes like this.
Mult the lot add 1, this new number is necessarily prime,
contradiction, proof.
I rehash this because I am thinking that perhaps these two variants
indicate that the very definition of primeness is amiss and that is why
we have a Hardy version and an AP version.
These are just some hunches. Perhaps if math definitions are 100%
good then the proofs concerning those definitions will provide only one
proof. So let me ask this question. I have no beef with any of the Real
Fundamental Theorem of Algebra. Is that proof of RFTA come in only one
flavor? Has the divergence of proofs of RFTA of Dalembert and Gauss
collapsed into one proof? But here I am straying away from IP.
My hunch. If the Adics had not 0 primes in them, but let us say 23
primes, just as a for instance. Then, it probably would never have come
to pass that we would have an Infinitude of Primes proof at all. For it
is obvious in some parts of math that 0 is the inverse of 00. And since
0 is the inverse of 00, then when we primitive practitioners of math
see that 2 and 3 look primey then a shoddy argument such as IP is
easily fobbed off as a 100% math proof. In other words, what I am
exploring is that I believe no primes exist at all from Adics and that
the reason that IP has an infinitude of primes is not because there are
an infinitude but because a fake argument was accepted. If IP leads to
many logically independent variants where one variant such as Hardy's
is incompatible with say my proof version, then it perhaps means that
the fault goes back to our primitive and foggy definition of primeness.
That is why I rehashed Peter Eckel's post for it seems that proofs of
IP have a strange element of "time" in it that a math proof should not
have.
I am just exploring here and not throwing any curves. All those
definitions you ask for Mr. Ng are the ones in current use.
Douglas J. Zare
>Eucl geom to Loba geom and vice versa. And you never posted it; you
>only posted irrelevant quiz questions saying that if I answered those
>offbeat questions you might out of some change of heart answer me
If you had solved them, I would have helped you. If you e-mail your
solutions to me, I will help correct them.
Are you sure it was a friend who gave you those conversions? Those
"conversions" were wrong. Check them.
My contention is that you do not understand any of the things you talk
about. The exercises were a way for you to learn more about the subject
and what you do not understand about the subject. I believe that this is
(educationally) more responsible than answering a misguided question by
someone who has neither the prerequisites nor a willingness to understand
the answer.
It seems to me that you dislike real mathematics and science in general
because they are unclear for you. You post parodies of mathematics and
science which only obstruct real discussions. I must conclude that this
is your intent, and these posts are actually attacks on the newsgroups.
>[...]
> And BTW, Zare, today I received an email by you which was routed to
^^^^^
The correct term is "cc".
>my sysadm. Your email is a complaint on me. [...]
>Are you Zare, the person directly responsible for the shortened
>lifespan of that post? That thread? [...]
If I were killing your posts, I wouldn't complain to your sysadmin. You
feign knowledge of Caltech. Refresh your memory on our Honor Code before
accusing me of taking advantage of you.
These are the maps someone gave AP.
>> >To map E -> R: theta{R} = theta{E}, r{R} = tan r{E}
>> >To map R -> E: theta{E} = theta{R}, r{E} = arctan r{R}
>> >To map E -> L: theta{L} = theta{E}, r{L} = tanh r{E}
>> >To map L -> E: theta{E} = theta{L}, r{E} = arctanh r{L}
> BTW, Zare, you are wrong about tan and arctan, just think about
>symmetry from E->R and E to L. [...]
The analogy between S^2 and H^2 is valid in many domains. *None* of these
maps preserve lines. Look up your high school trigonometry. In polar
coordinates, a line in E^2 not through the origin looks like
radius = k sec(theta+c) for some real constants k and c.
Spouting vague terminology may help in some areas, but it is quickly
exposed in mathematics. Perhaps you want to choose another area?
If not, look up the second homotopy or second homology group of R^2 and
figure out what it means about continuous maps from S^2 to R^2.
Douglas Zare
Archimedes Plutonium
Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:
Right here, right now I am going to tell the laypersons of math and
the NON-math people reading this thread what is going on concerning
Douglas Zare.
For over a week, perhaps 3 or 4 weeks Douglas has been plaguing my
threads with his obnoxious posts. Douglas for some reason thinks he
knows math, and for several reasons feels that I do not know math. And
so he has begun to plague my posts with his obnoxious dribble. Like
telling me I should get off of Internet and learn math from a Community
College and other assorted innuendos and dart throwing at me.
I am reminded here of the B movie "An Officer And A Gentleman" where
the sargent? hollers "I ..... CANNOT .... HEAR ..... YOU"
Well Douglas, baby, I cannot hear you. I see no counterexample.
What is my stereotype of Douglas Zare? It is this. Douglas I envision
was one of those High school and Uni students that wore bottle glass
thick glasses, and when invited to go to a beach party did so
reluctantly and then only if he could carry a calculus text to the
beach. Oh yes, Dougy desired the bathing beauties on the beach but he
could not find a way to engage them in a conversation on the
L'Hospital rule. Douglas in my opinion is a typical math nerd, the kind
that make a lot of math professors. They are good at regurgitation,
great at computation. But when it comes to MATH LOGIC, they slow down
to a grinding halt.
You see Douglas now realizes he is out of his league here. And that
is a pitiful shame and it goes to show that the Princeton graduate
students are far, far better than what ever Cal tech seems to offer up
to me as a math person to expose.
You see, Douglas is now realizing that he is an embarrassement to
math, and to math at Caltech. So, please tell us Dougy baby are you
just a Caltech graduate student or are you a Caltech professor? And
give us those counterexamples for now you are no longer and
regurgitation and computation turf. You are on my forte, math logic,
and, you have no chance of winning against me. I have already made
mincemeat out of your hobgoblin math illogic. Post your
counterexamples, DIMWIT Douglas.
Danny Calegari
>In article <3qp9ti$6...@gap.cco.caltech.edu>
>za...@cco.caltech.edu (Douglas J. Zare) writes:
>
>> >Hardy's A MATHEMATICIANS APOLOGY are wrong and invalid for they argue
>> >that there is a prime factor that may be missing. I argue in my proof
>> >that (P1 x P2 x ... x Pk) + 1 is necessarily prime.
>>
>
>What counterexamples? That number (P1 x P2 x ... x Pk) + 1 is
>necessarily prime in the indirect proof.
2*3*5*7*11*13+1 = 30031 = 59*509
Danny Calegari.
Danny Calegari
> [. . .]
> Hardy's proof went like this. Mult the lot add 1, either this number
>is prime or a prime factor not in original set exists, contradiction,
>proof. My proof, which says that Hardy's IP is flawed goes like this.
>Mult the lot add 1, this new number is necessarily prime,
>contradiction, proof.
Hardy's (Euclid's) proof need not actually be a proof by contradiction.
It is a constructive proof insofar as if we have n primes
p_1, p_2, p_3 . . . p_n we can construct an n+1 th prime, and therefore
by reiterating the process, arbitrarily many primes. It is a
constructive process, given p_1, p_2 . . p_n to produce
M=p_1*p_2* . . *p_n + 1
and it is a constructive process to factorize
M=q_1*q_2* . . *q_m.
The point is that the q_i and the p_j are distinct for any i,j.
M may or may not be prime for any particular set of p_1 . . p_n.
For p_1=2,p_2=3,p_3=5,p_4=7,p_5=11,p_6=13, we have
M=2*3*5*7*11*13+1=30031=59*509
> I rehash this because I am thinking that perhaps these two variants
>indicate that the very definition of primeness is amiss and that is why
>we have a Hardy version and an AP version.
> [. . .]
It is possible for a true statement to have a number of different
proofs.
>[. . .] For it
>is obvious in some parts of math that 0 is the inverse of 00.
I do not understand this statement. Perhaps you could clarify?
Regards,
Danny Calegari. :)
Douglas J. Zare
>multiply them and add 1, that new number is NECESSARILY prime.
The new number is also necessarily not prime. In the natural numbers you
get a contradiction. The type of contradiction is irrelevant. There are
many instances of several different proofs being given in popular
mathematics books; why does this one bother you?
Quickie: What are contradictions in the category of sets of mathematical
statements where morphisms are implications?
There are hundreds of correct proofs of the Pythagorean Theorem. They
are valid under the assumptions of Euclidean space. In the n-adics you do
not get a contradiction of any sort by assuming that there are only
finitely many primes. This is one of the differences between n-adics and
natural numbers. The n-adics are not well-ordered, but the natural
numbers are.
>[...]
>I am the
>first to uncover the illogic of so many popular renditions of Euclid's
>Infinitude of Primes indirect proof.
>[...]
There are plenty of true statements no one cares about. Why do you choose
a false statement to be the first to "prove"?
That mathematics is unclear and muddled in your mind is not a statement
about mathematics. You are stuck on the second page of a 100,000 page
tome, so you ridicule the authors and declare the work invalid. There are
more constructive attitudes.
Douglas Zare
Terry Tao
Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
>>t...@rugola.princeton.edu (Terry Tao) writes:
>>> Ah yes, but are Counting numbers = Adics? Show us a set with ...121212
>
>by asking him to define "counting" since counting is a foggy notion as
>per adics.
Oh, you don't know how to count? I'm sorry for you. Okay, well here's
a quick definition of the "counting" operation for you.
Take a set of objects and remove them one at a time. For each object
you remove, put a scratch on your cave wall. If you manage to remove
all the objects, then the scratches on your wall represents the count
of your set! If you can't finish removing all the objects, then the set
is infinite. Thus, for example, the count of the set {A,B,C} can be
written as |||.
A count of a finite set is called a "Counting number". The scratches
are just one way to describe this count. There are more fancy ways to
represent numbers, like the decimal system, or set-theoretic cardinals,
but I'll wait until you've sorted out your confusion over the
scratches-on-a-wall system. You can also do things like compare, add,
and multiply counting numbers, but that's getting a bit advanced.
Anyway, that's a brief introduction to the Counting numbers. Now, do
you think the counting numbers = the adics, Archie?
Terry
Patrick Reany
: In article <3qfnja$e...@news.xroads.com>,
: Patrick Reany <re...@xroads.com> wrote:
: >Now, DeLugt and I did not see eye-to-eye on what the 10-adics really
: >were. I wanted to treat them as purely formal and treat them as an
: >unordered, and without magnitude, set of formal entities. He disagreed and
: >I found out how easy it is to generate controvery over 10-adic integers.
: >The one thing we could agree on is that the four idempotents that he
: >discovered in the set are fascinating, and we couldn't understand how the
: >establishment in math couldn't find them interesting. Oh well.
: Well, the main reason is that these idempotents are trivial in the more
: useful representation of the 10-adics, which is as the direct sum of the
: 2-adics and the 5-adics. The idempotents are then (0,0), (0,1), (1,0),
: and (1,1).
I'm truly appreciative of the carefully expressed responses to my post on
the 10-adic integers, and this post by Terry is high quality. I do wish to
explain that the beauty of the 10-adic integers seems to me to lie in the
digital representation of them---that is, their hyperinteger formulation,
for it is there that they exhibit counter-intuitive properties. And it is
there that I wish to see them propagate as a sort of mathematical show
case to high schoolers. I'm not touting that the hyperintegers are better
for analysis, but they are better for low-level calculations with a
common calculator that can thrill high schoolers---at least that's my hope!
I wish to thank all who responded to my post on the 10-adic
integers, whether by posting or by email. I wish to followup on
these at this time.
When DeLugt first showed me his 10-adic integers, I was amazed,
but I kept bugging him to tell me their practical use. In time he
did and I mentioned them in my last post. I am convinced that
their most important practical use is to help people get
interested in mathematics. We tried back then to get standard
mathematics journal publishers interested in the hyperintegers,
but we were always told either that the subject is new and
researchy and thus not fit for a didactic journal, or that they
are well known result of g-adic theory and thus of no interest.
Of no interest to whom, I ask?!
I mentioned that I wanted to keep the hyperintegers a formal set,
meaning that I don't want to ascribe meaning to them in the
ordinary cardinal sense, such as one does with the ordinary
integers. However, I most certainly do not take a dispassionate
view to them. To me they are "magical" numbers, full of awe and
mystery.
I received responses that argued that from a purely formal view
the idempotents of the hyperintegers are "obvious" and thus not
so interesting. But, although they are correct as far as the math
goes, I cannot accept this attitude. First, because I am
committed to trying to get people interested in mathematics, the
most sublimely interesting invention of humanity (next to
refrigeration). And second, because there are many things in this
world of our experience that are obvious in some sense, but which
thrill us, or put us in awe.
Would you tell someone who never tasted sweet chocolate to not
bother tasting it because its sweetness is obviously derived from
sugar? Or would you tell someone from a region of endless plains
not to see the Grand Canyon because from a geological view the
G.C. is obviously just erosion? I say no. Isn't flight
aerodynamically obvious? I'm talking about the emotional appeal
of the hyperintegers, because their properties are so
counter-intuitive. I think many high-schoolers could get
interested in mathematics by being exposed to them. We'll never
know unless we try.
But I also got replies from people who agree with me that these
"infinite" idempotents are interesting, and that they have been
known to recreational mathematics for many years. (One said that
they were fully described in the Scientific American many years
ago. I've looked through its back issues to 1971, but I haven't
found it yet. Perhaps I missed it. Help finding it would be
greatly appreciated.) But I have mixed feelings to this response
too. Although I'm glad that others find hyperintegers
interesting, why then was I never exposed to them in my
highschool or undergraduate mathematics education? Why are
they--by intention or just indifference--kept such a secret from
the people would could gain the most from them?
Now let's get to some more theory. One post suggested that I show
the hyperintegers as expressible as the direct sum of two
principal ideals, and that is right. Let H be the hyperintegers.
Then we can form the principal ideals I_1 and I_2 by:
I_1 := H.e_1 and I_2 := H.e_2,
thus H = I_1 + I_2
I alluded to this in part in my last post when I talked about
projecting numbers onto the idempotents, and taking addition and
multiplication mod 10^n. We can do this systematically by
considering the set of all hyperintegers (or just integers) taken
mod 10^n, with modular addition and multiplication defined on
them. I call this resulting ring a _Truncation Ring of Order n_.
I envision that this ring is useful for computation, though I'm
no expert on computational theory.
Now, the idempotents generate two nontrivial square roots of
unity in the hyperintegers. Consider the number sigma defined by
sigma := e_1 - e_2
Thus, sigma^2 = (e_1 - e_2)^2 = e_1 - 2.e_1e_2 + e_2 = 1
So, in the hyperintegers unity has four square roots:
+-1,+-sigma. Fascinating! Well, to me anyway.
It was obvious to me that one approach that DeLugt could take in
his attempt at FLT was to show that only non-terminating
hyperintegers are solutions out of the set of hyperintegers. A
_terminating_ hyperinteger can be written as an ordinary integer,
that is, a finite string of digits with an infinite numbers of
contiguous zeros to the left. This is still a valid method toward
an alternative proof of the theorem.
Now to A.P.: It was obvious to DeLugt and me ten years ago that
FLT does not automatically apply over the set of hyperintegers,
not that it was designed to, mind you. But consider the typical
way the constraint placed on the integers x,y,z is often, though
not always, stated: that for integers x,y,z there is no integer n
larger than 2 such that
x^n + y^n = z^n
PROVIDED THE CONSTRAINT xyz <> 0. It's quite ironic that this
constraint--designed to eliminate the trivial cases in the
ordinary integers of solutions in which one or more of the
integers x,y,z are zero--also eliminates the idempotents of the
hyperintegers because they are mutually annihilating
zero-divisors of the hyperintegers. In other words, by keeping
the constraint xyz<>0, FLT may well in this form apply to the
hyperintegers!
One thing's for sure: the hyperintegers are NOT the Naturals, as
I have already said. That A.P. gets the two confused is not
because he's a crack pot. The very term "Naturals" is an atavism
from the naive time before Cantor's set theory, and it's just as
misleading as the name "imaginary" for the square root of -1. Now
one may rightfully disagree with me on this because there are,
after all, many views on the nature of mathematics itself.
However, if A.P. wants to better understand the 10-adics, he
should study-up on set theory and abstract algebra.
If we define "natural" in mathematics as something "true" that is
passively apprehended by human observation of Nature, then I
completely *disagree*. Everything in mathematics is a free
creation of the sentient mind and does not exist apart from this
active creation. (The alternative meaning of "natural," i.e., the
usual and/or canonical way that humans think of things, is
acceptable to me.) But if we use our conceptualization of a
finite set of physical object as our inspiration to defining an
abstract concept of number--called a Natural--that's OK. I'm not
sure how A.P. defines the Naturals; I'll define them informally
as the set of non-negative integers--that is the whole numbers,
and I do this because we will need the number 0.
I mention all this because sometime ago, A.P. asked the math
community to provide him with a rigorous definition of Natural
number, and I think it will be useful for high-schoolers to hear
it as well. The hyperintegers are a "natural" place for
high-schoolers to first deal with the subtle but interesting
aspect of infinity. There are a lot of good lay books on the
topic of introductory mathematical philosophy that go over
Peano's postulates to the formal definition of the
Naturals--Bertrand Russell's, _Introduction to Mathematical
Philosophy_ ---is one of them.
Now comes the real magic: It is necessary to distinguish the
notion of a particular natural number from the set of all natural
numbers. A.P. says that the (finite) Naturals as a set are
ill-defined. This is not so, especially for him. It might be so
for those that can't stomach the notion of an infinite set, but
this is not A.P.'s problem. Before Cantor's set theory, the
notion of a rigorous meaning to an infinite set was not accepted
generally, and there are still some who don't buy it. Cantor
argued that the notion of an unbounded number sequence
presupposed the notion of an infinite set to draw them from. But
A.P.'s approach is just the reverse. He contends that instead of
only finite sets existing, rather, only infinite sets exists. Now
this _is_ bizarre. The problem with this approach is that the
hyperintegers require the concept of a natural (finite) number
just to define them. Then it is again required to justify that
the hyperintegers form a ring under addition and
multiplication--that is, we say that a hyperinteger sum or
product is well defined if given any natural number n then the n
rightmost digits of the hyperinteger can in principle be
calculated in a finite number of operations. Whether not or this
is enough for one to accept that these operations conform to ring
closure is an act of faith--an act I am willing to take.
Thus the hyperintegers offer a "natural" place for a high-school
student to begin his or her search for understanding infinity,
set theory, rings and ideals, abstract algebras, and so on, but
with true motivation from within!
If we're ever going to engage a larger percent of the population
to take up mathematics, even as a hobby, we have to expose high
schoolers to more than the quadratic formula and Cramer's rule,
important as they are. If they come to college hating
mathematics, or just thinking that it's interminably boring, how
will they ever know the wonders of distribution theory and
Green's functions, of topology and Clifford algebras, of set
theory and category theory, and all the rest? One way to get them
started is through exposure to the hyperintegers.
I'd like to thank Santi for explaining to A.P. that I'm not
trying to jump his claim to fame. For all I care, let Kummer,
Hensel, DeLugt, A.P. and all the rest get all the glory.
There are a number of references now available on p-adics:
_Algebra_ by Serge Lang gives it a brief mention. _A Course in
Number Theory_, by H.E.Rose. _p-adic Analysis_, by Neal Koblitz
(Cambridge U. Press). _p-adic Numbers, ..._, by Neal Koblitz
(Springer-Verlag). _p-adic Analysis and Mathematical Physics_, by
Vladimirov, et al (World Scientific Publishing). (But don't
expect much, if any, mention of 10-adics in these sources.)
cheers
-- Patrick <re...@xroads.com>
Dik T. Winter
> > 2*3*5*7*11*13+1 = 30031 = 59*509
>
> primes finite. Then you can list them. As in your case it is this
> {2,3,5,7,11,13}. Since you can list them they have a largest, ie, 13.
> Multiply them all and add 1 which is 30031. That number is necessarily
> prime by the unique prime factorization theorem because if all the
> primes are finite then all of them are in that existing list and none
> of them factors into 30031 so it is also prime. Contradiction.
Or: That number is necessarily not prime because if all the primes are
finite then all of them are in that existing list and none of them
equals 30031. But because of the unique prime factorization theorem
30031 should factor in primes and none of the existing primes divide
it. Contradiction.
Both reasonings are equally valid. Going from the premissa that you have
a complete list of all primes you can conclude that the product of all of
them with 1 added is either:
1. not prime because it is not in the list,
or
2. prime because it does not factor with the primes given.
Both lead to a contradiction.
John Chandler
Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
>In article <3qs5oq$3...@dartvax.dartmouth.edu>
>Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:
>
>> za...@cco.caltech.edu (Douglas J. Zare) writes:
>>
>> > >Hardy's A MATHEMATICIANS APOLOGY are wrong and invalid for they argue
>> > >that there is a prime factor that may be missing. I argue in my proof
>> > >that (P1 x P2 x ... x Pk) + 1 is necessarily prime.
>> >
>> > Did you not understand/believe the counterexamples?
>>
>> What counterexamples? That number (P1 x P2 x ... x Pk) + 1 is
>> necessarily prime in the indirect proof.
The primes are 2, 3, 5, 7, 11, ...
Do you claim that one plus the product of the first k primes
is necessarily prime, for any integer k>0?
Yes or no, A.P., yes or no?
--
John Chandler
j...@a.cs.okstate.edu
Archimedes Plutonium
t...@yam.princeton.edu (Terry Tao) writes:
> A count of a finite set is called a "Counting number". The scratches
> are just one way to describe this count. There are more fancy ways to
> represent numbers, like the decimal system, or set-theoretic cardinals,
> but I'll wait until you've sorted out your confusion over the
> scratches-on-a-wall system. You can also do things like compare, add,
> and multiply counting numbers, but that's getting a bit advanced.
>
> Anyway, that's a brief introduction to the Counting numbers. Now, do
> you think the counting numbers = the adics, Archie?
Counting is well defined for a finite set because it can be associated
with the finite portion of Reals. Not the infinite string rightwards
with Reals, nor with any infinite string.
To give you some analogies. In physics, matter has properties which
make sense only in the macroscopic level and no sense in the atomic
level, such as color or friction.
No, I never thought Counting numbers = Adics. Counting is not an
instrinsic math DNA or RNA. Counting is one of humanities subjective
feelings injected onto math, just as absolute space and time and
causality are other human animations which do not exist. But we will
clear these animations out via pragmatism and science experiments.
I have maintained constantly the famous equation Naturals = Adics =
Infinite Integers. Pluto must be put in a near Earth orbit, sorry, a
touch of Abian just hit me.
Archimedes Plutonium
Danny Calegari <dan...@mundoe.maths.mu.oz.au> writes:
> 2*3*5*7*11*13+1 = 30031 = 59*509
Danny I do not know why you defend Douglas Zare. His arrogance is too
much to stomach, but I suppose the above is what he would list. Notice
he has shyed away now.
No that is not a counterexample to the proof of IP. Suppose all
primes finite. Then you can list them. As in your case it is this
{2,3,5,7,11,13}. Since you can list them they have a largest, ie, 13.
Multiply them all and add 1 which is 30031. That number is necessarily
prime by the unique prime factorization theorem because if all the
primes are finite then all of them are in that existing list and none
of them factors into 30031 so it is also prime. Contradiction. Both 59
and 509 are not primes of all existing finite primes because that was
in the supposition.
This is what is so pathetically wrong with virtually every proof
attempt of Euclid's Infinitude of Primes. Mult the lot and add 1, by
pure logical reasoning is necessarily prime. When Hardy or Montgomery &
Niven & Zuckerman look for counterexamples they have abandoned logic,
they have abandoned the earlier steps that suppose finite then they can
be listed and those are the only existing primes. They have abandoned
the existential quantifier of a good proof and have fallen into a
flawed argument. All indirect proofs of IP force that new number as
necessarily prime.
The only problem I have with it is that Naturals = Adics = Infinite
Integers means really that no primes exist at all. I am trying to
fathom a better definition of primeness. I believe it will come from
the Whole Number Reals such as these 2.000.... , 3.00000.... ,
4.0000.... and that you can only talk about primeness within a finite
set. It might be that within any infinite set, primeness is an illusion.
Archimedes Plutonium
> Hardy's (Euclid's) proof need not actually be a proof by contradiction.
> It is a constructive proof insofar as if we have n primes
> p_1, p_2, p_3 . . . p_n we can construct an n+1 th prime, and therefore
> by reiterating the process, arbitrarily many primes. It is a
> constructive process, given p_1, p_2 . . p_n to produce
>
> M=p_1*p_2* . . *p_n + 1
Yes, concur and this is the Direct proof method which I posted a long
time ago. Basically it is the same type of proof that proves Naturals
are infinite. Proof: No matter where you stop you add 1 to last member
to increase cardinality of set, or, any finite set has a largest member
add 1 to increase cardinality.
But here we see the flaw in this type of proof of infinity in regards
to Naturals = Adics for there really are no largest integer of a given
set of adics.
No, what I was trying to pick apart with the usual flawed reasoned
indirect proof of say Hardy or Montgomery & Niven & Zuckerman and my
own version is that it seems as though all proofs of IP are inelegant
and rather dirty with gaps. As Peter Eckel noted that IP seems to have
a time function within the proof, that at some point of the proof you
want to scrutinize this new number and go to a contradiction but that
part of the proof does not flow beautifully like a real math proof if
true would flow. A beautiful proof like the pythagorean theorem.
And there have been many proofs of IP in various other math subjects
such as topology. And I wonder if they have the same pitfalls?
Archimedes Plutonium
za...@cco.caltech.edu (Douglas J. Zare) writes:
> The new number is also necessarily not prime. In the natural numbers you
> get a contradiction. The type of contradiction is irrelevant. There are
> many instances of several different proofs being given in popular
> mathematics books; why does this one bother you?
False. A rigorous proof would use definitions of prime and composite
(>1). All proofs of IP use Unique Prime Factorization (UPF). When you
form the new number mult the lot add 1, the UPF guarantees it is prime.
Of course when you want to write up a proof and disregard half of the
proof you can say almost anything about the new number, which is just
another sloppy invalid proof like Hardy's or Montgomery,Niven,
Zuckerman.
ningauble (of the seven eyes)
> Pluto must be put in a near Earth orbit, sorry, a
> touch of Abian just hit me.
huh? would you care to explain a little more thoroughly?
--b.d.s.
John Chandler
Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
[munch]
> Hardy's proof went like this. Mult the lot add 1, either this number
>is prime or a prime factor not in original set exists, contradiction,
>proof. My proof, which says that Hardy's IP is flawed goes like this.
>Mult the lot add 1, this new number is necessarily prime,
>contradiction, proof.
Are you saying that if you multiply the first n primes together
and add one, the result is ALWAYS prime?
Not true at all, of course!
This is a common misconception of beginning students.
(Do you know the first counterexample to this?
Why not show it to us,
so that we will know that you can at least
solve very elementary problems?)
That is why it is absolutely necessary to add the possibility that
"a prime factor not in the original set exists",
and why any proof without that possibility is flawed.
--
John Chandler
j...@a.cs.okstate.edu
Matthew Blair
Plutie is refering to Alexander Abian, a sci.physics poster. He apparently
is quite the accomplished mathematician; however, his current passion
appears to be the advancement of his Abian TIME-MASS EQUIVALENCE. This
equivalence claims that mass is consumed to generate time. Oh yeah, in
reference to what Pluto said, Abian's sig contains a phrase to the effect
that Venus must be put in a near earth orbit or something like that.
Check it out. It makes for quite amusing reading...
Matthew
--
Matthew Blair | Usual disclaimers apply
Purdue University | #include <disclaimers.h>
EE student | This space for rent
Internet address: ma...@expert.cc.purdue.edu
Carsten Witzel
: Could somebody recommend a good book that defines -adics and discusses
: them in an elementary way?
I'd suggest
Gouvea, Fernando Q.: P-adic numbers: an introduction; Springer, 1993,
or
Koblitz, Neal: P-adic numbers, p-adic analysis, and zeta-functions;
Springer, 1984.
--
Carsten Witzel -- E-Mail: wit...@uni-duesseldorf.de
Archimedes Plutonium
l...@panix.com writes:
> It seems to me that you just cannot escape from counting.
I do not dispense with counting. I make the definition more precise
than what is currently in usage. The definition comes out of Reals with
the whole numbers in Reals where 94.0000... and 231.00000... are there.
So make a more precise definition of Counting and of Math Induction
from Reals, but remember the leftward string is finite.
Archimedes Plutonium
d...@cwi.nl (Dik T. Winter) writes:
> Or: That number is necessarily not prime because if all the primes are
> finite then all of them are in that existing list and none of them
> equals 30031. But because of the unique prime factorization theorem
> 30031 should factor in primes and none of the existing primes divide
> it. Contradiction.
>
> Both reasonings are equally valid. Going from the premissa that you have
> a complete list of all primes you can conclude that the product of all of
> them with 1 added is either:
> 1. not prime because it is not in the list,
> or
> 2. prime because it does not factor with the primes given.
> Both lead to a contradiction.
I disagree. It sounds as if your moment of contradiction is a
floating quantifier. As in your 1. not prime because it is not in the
list you can apply the moment of contradiction at anyway through the
proof and why even bother with multiplying the lot why not just add one
to the largest member and since it is not in the list, you can say
contradiction.
No, I disagree because I am convinced that the extraction of another
prime not in the finite list is the moment you can raise the issue of
contradiction. There, another prime. But all the primes were listed,
hence contradiction.
This is what I mean about the seeming time factor involved in IP.
l...@panix.com
returned from your vacation from the Internet. Welcome back
(belatedly). :)
In article <3qvvan$g...@dartvax.dartmouth.edu>, Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:
> In article <3qtpuo$7...@cnn.Princeton.EDU>
> t...@yam.princeton.edu (Terry Tao) writes:
>
> > A count of a finite set is called a "Counting number". [ ... ]
> >
> > Anyway, that's a brief introduction to the Counting numbers. Now, do
> > you think the counting numbers = the adics, Archie?
>
> [ ... ]
>
> No, I never thought Counting numbers = Adics. Counting is not an
> instrinsic math DNA or RNA. Counting is one of humanities subjective
> feelings injected onto math, just as absolute space and time and
> causality are other human animations which do not exist. But we will
> clear these animations out via pragmatism and science experiments.
If I may interject here ...
In my humble opinion, counting is indeed a very basic part of "math
DNA or RNA". The concept of counting is even built right into FLT.
Consider this:
A^N + B^N = C^N
In English, this translates to "A to the N-th power plus B to the N-th
power equals C to the N-th power". Now, what does "A to the N-th
power" really mean? In my high-school-math-teacher level of
understanding, it means "multiply A by itself N times". This relies
at a very basic level on the concept of *counting* ... i.e., N is the
*count* of the number of times A is multiplied by itself ... we
*count* the number of A's that are multiplied, and that is what N is.
As long as N is an integer, this concept of counting is therefore
intrinsic in the concept of raising a number to the N-th power.
You often state that "Naturals = Adics = Infinite Integers". You also
stated above that "No, I never thought Counting numbers = Adics." On
the basis of these two statements, it can be easily inferred that you
do not consider "Counting Numbers" to be the same as "Naturals" (or
"Adics", or "Infinite Integers").
But Fermat was specifically referring to "Counting Numbers" when he
formulated FLT. He was specifically talking about a theorem that
operates solely on counting numbers. Again, consider this formula:
A^N + B^N = C^N
When Fermat formulated his theorem concerning this equation, he was
referring to A, B, C, and N all being Counting Numbers, which you have
stated here are not the same as Adics. At that time in history,
people referred to Counting Numbers as "Naturals".
You have changed the definition of "Naturals" to mean "Adics", but
that doesn't change the fact that Fermat was dealing with Counting
Numbers, and that Adics are not Counting Numbers.
If A, B, C, and N in the formula above are meant to stand for "Adics"
instead of "Counting Numbers", then you are not dealing with "FLT" at
all. This is because "FLT" stands for "Fermat's Last Theorem", and
because Fermat himself was dealing with Counting Numbers and not
Adics. You have a completely different theorem to prove or disprove
if you define A, B, C, and N to be Adics and not Counting Numbers. It
is "false advertising" (in my respectful opinion) to use the name
"Fermat's Last Theorem" (or the abbreviation "FLT") for this
Adic-based theorem, because (and I repeat for emphasis) Fermat
formulated his "Last Theorem" using Counting Numbers and not Adics.
Perhaps a better name for this Adic-based version of the theorem would
be "Plutonium's First Theorem" ("PFT"), or something similar. But
whatever name is chosen, let's not confuse your new theorem with
Fermat's older theorem.
It's also noteworthy that "PFT" is much easier to prove/disprove than
"FLT", since Adics have some different properties than Counting
Numbers.
And someone correct me if I'm wrong: isn't it true that Wiles is
working on the traditional FLT (the one that uses Counting Numbers),
and not the other, newly created "PFT" that looks kind of like FLT,
but is really something different (since it uses Adics and not
Counting Numbers)?
> I have maintained constantly the famous equation Naturals = Adics =
> Infinite Integers. [ ... ]
You have redefined "Naturals" to be something other than "Counting
Numbers". That is your prerogative. Just don't forget that if you
choose to stop working with Counting Numbers, then you are no longer
dealing with Fermat's original (and very hard to prove/disprove)
theorem, since that theorem is all about Counting Numbers.
And one more key point ... once again, consider this formula:
A^N + B^N = C^N
In "PFT", you show a counter-example to Fermat's assertion using (if
my memory serves me), 5-Adics. However, it is my belief (and please
correct me if I am wrong), that in your counter-example, only A, B,
and C are really Adics. The "N" is actually a Counting Number because
it seems to me (and again, correct me if I am wrong) that raising an
Adic to the N-th power has exactly the same meaning as raising a
Counting Number to the N-th power: i.e., you multiply the Adic by
itself N times, which I have already shown above is an operation that
depends on N being a Counting Number.
So ... even in "PFT" (as opposed to its older cousin "FLT"), it seems
that you cannot escape from Counting Numbers.
And one last question, Archimedes, respectfully asked:
You made the following statement (above) ...
> [ ... ] Counting is one of humanities subjective feelings injected
> onto math, just as absolute space and time and causality are other
> human animations which do not exist. But we will clear these animations
> out via pragmatism and science experiments.
It sounds like you are saying that counting is something that will not
stand up to the test of pragmatic experimentation. If that is the
case, then on what basis to you deal with the investment strategies
you pursue? Isn't it true that calculating the amount to invest and
the return on your investments involves lots of *counting*?
Doesn't the fact that you pursue investment at all prove that you
consider the concept of *counting* to be a very basic, pragmatic,
and experimentally sound practice?
And isn't it true that when you are talking about the possibility of
currency collapse in the USA, you are actually talking about a process
that makes deeply intrinsic use of the concept of *counting* (i.e.,
the *count* of how much currency is available, the *count* of how many
dollars it takes to buy a bicycle now versus after the currency
collapse, etc.)?
It seems to me that you just cannot escape from counting.
94 Pu 137
^ ^
| |
+-----+----- these are Counting Numbers
--
Lloyd Zusman 01234567 <-- The world famous Indent-o-Meter.
l...@panix.com ^ I indent thee.
To get my PGP public key automatically mailed to you, please
send me email with the following string as the subject or on a
line by itself in the message (leave off the quotation marks):
"mail-request public-key"
Dik T. Winter
> All proofs of IP use Unique Prime Factorization (UPF).
No, none of them use that the factorization is unique; they only use that
it does exist. The proof also holds for the Gaussian integers that do not
have UPF. (They also use that primes are not units, but that is by
definition.)
> When you
> form the new number mult the lot add 1, the UPF guarantees it is prime.
No, it guarantees that it is factorizable in primes. That it is prime is
already a conclusion. See:
Define: P = finite set of all primes, M is number created; assume UPF.
{M not in P} -> {M is not prime}
{M does not factorize} -> {M is prime}.
{M is prime} + {M is not prime} -> contradiction.
You start with the second, others start with the first. But both are
equally valid.
Archimedes Plutonium
> Both reasonings are equally valid. Going from the premissa that you have
> a complete list of all primes you can conclude that the product of all of
> them with 1 added is either:
> 1. not prime because it is not in the list,
> or
No, because you cannot get a contradiction. All primes existing are in
the finite list. Find one more then it contradicts the "all existing
primes".
Archimedes Plutonium
d...@cwi.nl (Dik T. Winter) writes:
> No, none of them use that the factorization is unique; they only use that
> it does exist.
True, I just call it that theorem, call it Unique Prime Factorization
Theorem. That name is alot prettier to me than Fund. Theorem of
Arithmetic or some other name. I just use the name, not attaching
significance to the "unique" part.
Archimedes Plutonium
> No, it guarantees that it is factorizable in primes. That it is prime is
> already a conclusion. See:
Concur.
> Define: P = finite set of all primes, M is number created; assume UPF.
> {M not in P} -> {M is not prime}
This cannot be arrived at. Only M is prime can be arrived at because
you know a definition of prime at the outset of the proof--- a
definition such as is factorable by some existing primes.
Dik T. Winter
> In article <D9q4y...@cwi.nl>...
> > Or: That number is necessarily not prime because if all the primes are
> > finite then all of them are in that existing list and none of them
> > equals 30031. But because of the unique prime factorization theorem
> > 30031 should factor in primes and none of the existing primes divide
> > it. Contradiction.
> As in your 1. not prime because it is not in the
> list you can apply the moment of contradiction at anyway through the
> proof
> and why even bother with multiplying the lot why not just add one
> to the largest member and since it is not in the list, you can say
> contradiction.
it is not prime. But that does not lead to a contradiction.
But pray tell, what part of the above proof is invalid?
...
> No, I disagree because I am convinced that the extraction of another
> prime not in the finite list is the moment you can raise the issue of
> contradiction. There, another prime. But all the primes were listed,
> hence contradiction.
Why? If starting from a premissa, using a further valid proof, I can
conclude two contradicting things, I know the premissa was wrong.
You can raise the issue of contradiction the moment you get at a
contradiction. In my proof I found that M is divisible by at least
one member of P, but I also found that it is not. That is a contradiction.
Zdislav V. Kovarik
Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
:>In article <3r07po$p...@news.panix.com>
:>l...@panix.com writes:
:>
:>> It seems to me that you just cannot escape from counting.
:>
:> I do not dispense with counting. I make the definition more precise
:>than what is currently in usage. The definition comes out of Reals with
:>the whole numbers in Reals where 94.0000... and 231.00000... are there.
:>So make a more precise definition of Counting and of Math Induction
:>from Reals, but remember the leftward string is finite.
Did everyone notice the shifting of responsibility for definitions and
proofs form A.P to his/her/its/their opponents? (Not to mention the
clowning with numbers 94 and 231). A.P. is simply baiting you, and has
acquired such a mastery in this that he/she/it/they can be awarded the
degree of Master Baiter.
Have fun, ZVK (Slavek).
John Greene
> Both reasonings are equally valid. Going from the premissa that you have
> a complete list of all primes you can conclude that the product of all of
> them with 1 added is either:
> 1. not prime because it is not in the list,
> or
> 2. prime because it does not factor with the primes given.
> Both lead to a contradiction.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924098
> home: bovenover 215, 1025 jn amsterdam, nederland; e-mail: d...@cwi.nl
I can think of at least two other possibilities!
3. The product + 1 is not prime because it is a unit. This is what happens
in the 10-adics when we calculate (2)(5) + 1 = 11 which is not a prime
in the 10-adics.
4. The product + 1 is not prime because it factors as a product of two
non-units. This is why I find AP's proof asthetically distasteful-
in the "counterexample" (2)(3)(5)(7)(11)(13) + 1 = 30031 = (59)(509),
what is to stop one from saying that this is a factorization into
non-units (ignore the question of whether 59 and 509 are primes)
so the number in question is not prime?
With regard to (4), I've told AP before that if he wants to maintain that
his proof is best, he first has to prove the following: If n > 1 and n
is not divisible by a PRIME less than it, then n is prime.
This is, of course, true for what AP calls the "old fashion" natural numbers
or the "real primes" but I've never seen him prove it or even acknowledge
this point.
John
Douglas J. Zare
>time ago. Basically it is the same type of proof that proves Naturals
>are infinite. Proof: No matter where you stop you add 1 to last member
>to increase cardinality of set, or, any finite set has a largest member
>add 1 to increase cardinality.
Before, you claimed that the cardinality of the naturals is finite. You
also claimed that the cardinality of the reals is finite.
>But here we see the flaw in this type of proof of infinity in regards
>to Naturals = Adics for there really are no largest integer of a given
>set of adics.
What was this sentence supposed to mean? If you post to hundreds of
servers around the world, you should express something.
>[...]
>As Peter Eckel noted that IP seems to have
>a time function within the proof, that at some point of the proof you
>want to scrutinize this new number and go to a contradiction but that
>part of the proof does not flow beautifully like a real math proof if
>true would flow. A beautiful proof like the pythagorean theorem.
>[...]
Words which suggest time are almost always merely a matter of
convenience or presentation style. Proofs can be transformed logically so
as to avoid such words. If those words bother you then do so; the
aesthetics are irrelevant to the mathematical content.
Douglas Zare
Archimedes Plutonium
j...@a.cs.okstate.edu (John Chandler) writes:
> The primes are 2, 3, 5, 7, 11, ...
> Do you claim that one plus the product of the first k primes
> is necessarily prime, for any integer k>0?
>
> Yes or no, A.P., yes or no?
>
>
> --
> John Chandler
IF the first k primes are all the primes that exist (NOTE THAT- all
the primes that exist) (as in the reductio ad absurdum proof), then the
answer is YES, YES, YES.
It really is a shame that most people in math do not have the math
logic to spit into a bucket. And math logic is not something you can
teach for it is ingrained in someones bones from birth. That is a
quote from a dear old gentleman here in sci.math.
Dik T. Winter
> In article <D9q5r...@cwi.nl>
> > {M not in P} -> {M is not prime}
> This cannot be arrived at.
Why can I not conclude that given P {the complete list of primes} and
M is not in P, that M is not prime?
> Only M is prime can be arrived at because
> you know a definition of prime at the outset of the proof
This is irrelevant. I am given a complete list, M is not in the list,
so M does not have the property. Done!
David Empey
Archimedes...@dartmouth.edu (Archimedes Plutonium) wrote:
> In article <3qv7br$h...@bubba.ucc.okstate.edu>
> j...@a.cs.okstate.edu (John Chandler) writes:
>
> > The primes are 2, 3, 5, 7, 11, ...
> > Do you claim that one plus the product of the first k primes
> > is necessarily prime, for any integer k>0?
> >
> > Yes or no, A.P., yes or no?
> >
> >
> > --
> > John Chandler
>
> IF the first k primes are all the primes that exist (NOTE THAT- all
> the primes that exist) (as in the reductio ad absurdum proof), then the
> answer is YES, YES, YES.
Well, fools jump into the middle of threads where wiser men keep silent,
but I just can't resist:
If the first k primes are ALL the primes that exist, how can any other number
(such as the product of the first k primes, plus 1) be a prime?
--Dave Empey (dge...@cats.ucsc.edu)
Jesus didn't even have a utility belt! How lame can you get?
l...@panix.com
> In article <3r2r9r$7...@dartvax.dartmouth.edu>,
> Archimedes...@dartmouth.edu (Archimedes Plutonium) wrote:
>
> > In article <3qv7br$h...@bubba.ucc.okstate.edu>
> >
> >
> > IF the first k primes are all the primes that exist (NOTE THAT- all
> > the primes that exist) (as in the reductio ad absurdum proof), then the
> > answer is YES, YES, YES.
>
> Well, fools jump into the middle of threads where wiser men keep silent,
> but I just can't resist:
>
> If the first k primes are ALL the primes that exist, how can any other number
> (such as the product of the first k primes, plus 1) be a prime?
You are absolutely right on track! You have hit upon the essence of
the "indirect proof" method:
(1) Make an initial assumption.
(2) Derive some results based upon this initial assumption.
(3) Show that one or more of your results contradicts this initial
assumption.
(4) This proves that your initial assumption was false.
This is often used to prove the truth of an assertion, as follows:
(1) Make an initial assumption that is the inverse of your assertion.
(2) Derive some results based upon this inverse assumption.
(3) Show that one or more of your results contradicts this inverse
assumption.
(4) Since this proves that your inverse assumption was false, it proves
that your original assertion is true.
Let's apply these 4 steps to the "infinitude of primes" problem:
ORIGINAL ASSERTION: There are an infinite number of primes.
STEP (1) -- Create an inverse assumption:
INVERSE ASSUMPTION: There are only a finite number of primes.
STEP (2) -- Derive some results based upon this inverse assumption:
Since there are a finite number of primes, the following set represents
all of these primes:
(P1, P2, P3, ..., Pn)
... where 'n' is a finite number (an aside to AP: note that 'n' is a
*Counting Number* !)
Call this set Q.
Now, calculate the result X, which is the product of all these values,
and with the number 1 added:
X = (P1 x P2 x P3 x ... x Pn) + 1
Because of how X is derived above, if you divide X by any of the
primes in set Q (P1, P2, P3, ..., Pn), you will get a remainder of 1.
DERIVED RESULT: There is at least one value, X, which is not evenly
divisible by any of the primes in our initial set.
STEP (3) -- Show that one or more of your results contradicts this inverse
assumption.
Well, our "INVERSE ASSUMPTION" is that the set of primes is finite. We
applied addition and multiplication to all the numbers in this set,
and we calculated another number, X. Our "DERIVED RESULT" is that X
is not a part of our set of primes, and that X is not evenly divisible
by any of the numbers in our set of primes. If a number is not
evenly divisible by any prime (except itself), then it is by definition
a prime.
This shows that our "DERIVED RESULT" contradicts our "INITIAL
ASSUMPTION", since we stated that set Q is the set of *all* primes,
but we have, using this as an assumption, derived a prime that is not
in set Q.
STEP (4) -- Since this proves that your inverse assumption was false,
it proves that your original assertion is true.
The contradiction in STEP (3) shows that our INVERSE ASSUMPTION (i.e.,
that the set of primes is finite) is false. This therfore proves that
our ORIGINAL ASSERTION (that the set of primes is infinite) is true.
This is the indirect proof of "the infinitude of primes" that is being
discussed here, and the contradiction you mentioned in your post is
the key to performing this proof.
Alan Morgan
Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
>In article <3qv7br$h...@bubba.ucc.okstate.edu>
>j...@a.cs.okstate.edu (John Chandler) writes:
>
>> The primes are 2, 3, 5, 7, 11, ...
>> Do you claim that one plus the product of the first k primes
>> is necessarily prime, for any integer k>0?
>>
>> Yes or no, A.P., yes or no?
>
>the primes that exist) (as in the reductio ad absurdum proof), then the
>answer is YES, YES, YES.
Folks, he's right. If you assume the first k primes are all that exists
then their product plus 1 must equal a prime. It is also the case that
if you assume this, that 2 is odd, 3 is greater than 4, and I am bright
pink.
It is very easy though, to fall into the trap of believe that this method
allows up to construct new prime numbers. It doesn't.
Of course, AP has got no right to get snotty about this since he misuses
this same proof technique when "proving" that there are an infinite
number of paired primes.
Alan
Archimedes Plutonium
l...@panix.com writes:
> This is the indirect proof of "the infinitude of primes" that is being
> discussed here, and the contradiction you mentioned in your post is
> the key to performing this proof.
Good on you Lloyd. And I was saying to myself before opening up the
Internet today that I hope I do not see 50 posts by people that I have
to straighten out.
Good proof Lloyd. Yours is valid. Hardy's, Ian Stewart's,
Montgomery&Niven& Zuckerman 's, Dunham's, Paulos's are all flawed and
invalid attempts. And I could name a score more of invalid attempts
that are written up in math texts or books. It is all so laughable. For
as one math professor here at Dartmouth once told me. No math person is
worth their weight in salt in math if they cannot give you a valid
proof of Euclid IP. That is true, and so 99% of the people engaged in
math probably fail the simplest and easiest of math proof gems. Read
Hardy's A MATHEMATICIAN'S APOLOGY on this proof where he waxs
poetically calling it a veritable diamond of a proof and says not a
wrinkle has been written into it since Euclid. Well, I am sorry to say
but noone before 1993 could even give a valid proof of it.
But I cannot get excited about it for in 1993 I discovered Naturals =
Adics = Infinite Integers and no primes exist at all within Adics. So I
cannot get too excited over an argument that says an infinitude of
primes exists. Can I?
I think the resolution will come out of Reals. That there exists
Whole numbers in Reals and the adics are Whole numbers. Such as 2.00...
Real is the adic ....00002 What we discard onto the junkpile as foggy
notions is that of finite and finite integer.
Archimedes Plutonium
d...@cwi.nl (Dik T. Winter) writes:
> In article <3r2lhn$p...@dartvax.dartmouth.edu> Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:
> > In article <D9q5r...@cwi.nl>
> > d...@cwi.nl (Dik T. Winter) writes:
> > > Define: P = finite set of all primes, M is number created; assume UPF.
> > > {M not in P} -> {M is not prime}
> > This cannot be arrived at.
>
> Why can I not conclude that given P {the complete list of primes} and
> M is not in P, that M is not prime?
You cannot do that because in order to get a contradiction you must
produce a new prime not on the list of all primes. Not until you pick
up a new prime can you say "contradiction". You can say contradiction
once you have found a new prime not on that list of All primes. You
can say contradiction then, because obviously it is not All primes.
Say you accept M as not prime because of your earlier steps saying
that you had the complete list of the primes. What are you going to do
next Dik? You still need to produce a prime not on that list of all
primes. Only by producing a new prime not on the list of all primes can
you ever utter "contradiction".
>
> > Only M is prime can be arrived at because
> > you know a definition of prime at the outset of the proof
>
> This is irrelevant. I am given a complete list, M is not in the list,
> so M does not have the property. Done!
It is not irrelevant. The Fundamental Theorem of Arithmetic (UPF) is
completely connected with this indirect proof. UPF gives that the two
new numbers W-1 and W+1 are Necessarily prime.
You are not done. You have not produced a new prime.
Here is the proof in a paragraph. Proof. Suppose false. Then the
primes can be listed with a largest prime P_L. Multiply the lot add 1,
call it W+1. W+1 is larger than P_L. W+1 is necessarily prime since by
UPF, all the primes that exist leave a remainder of 1 and by definition
of prime. W+1 is not a member of all the primes that exist.
Contradiction. Therefore true that primes are infinite.
(BTW, some viewers have emailed me saying I must say something about
W+1 is not equal to 1. This is incorrect for we know all along that
every prime is larger than 1 and we know multiplication of greater
than, so therefore W+1 is larger than 1. This 1 business has no need to
be mentioned in a proof of IP.)
Douglas J. Zare
>but noone before 1993 could even give a valid proof of it.
There are many, many proofs that there infinitely many primes. You have
become hung up on the fact that you see a slightly different
contradiction than what is stated in the elementary texts and popular
mathematics books you read. It is a shame you do not understand that
contradictions are equivalent (initial objects in the category I
mentioned of sets of mathematical assertions with implications as
morphisms, btw).
Here is another standard proof which is slightly above the one you gave.
The harmonic series diverges, yet it can be written as
1
Product -------
prime p 1-(1/p)
Oviously each term of the product is finite, so if there are finitely
many primes, then the harmonic sum would converge. It diverges, so there
must be infinitely many primes.
Another way of saying this is that for any finite set of primes, the
density of the set of positive integers divisible only by the elements of
that set is 0. The natural numbers have density 1, so there are
infinitely many primes.
You ridiculed Erdos and others. This demonstrates that your ignorance. In
fact, one early theorem of Erdos (was it when he was a teenager?) was
that there exists a prime between n and 2n for every natural n.
> But I cannot get excited about it for in 1993 I discovered Naturals =
>Adics = Infinite Integers and no primes exist at all within Adics. So I
>cannot get too excited over an argument that says an infinitude of
>primes exists. Can I?
2 is prime in the 10-adic integers. 2 is not invertible and it is not the
product of 10-adic integers, neither of which is invertible.
> I think the resolution will come out of Reals. That there exists
>Whole numbers in Reals and the adics are Whole numbers. Such as 2.00...
>Real is the adic ....00002 What we discard onto the junkpile as foggy
>notions is that of finite and finite integer.
It is a shame that they are still foggy for you. After all this work, will
you discover the canonical embedding of the naturals in the reals?
In these posts, you are merely showing your difficulty understanding
elementary mathematics, your unwillingness/inability to think, and your
dislike and disrespect for those who can do mathematics. Now that we know
those, please stop posting about them to sci.math.
Douglas Zare
Patrick Reany
: In article <3qs5oq$3...@dartvax.dartmouth.edu>
: Archimedes...@dartmouth.edu (Archimedes Plutonium) writes:
: > In article <3qp9ti$6...@gap.cco.caltech.edu>
: > za...@cco.caltech.edu (Douglas J. Zare) writes:
: >
:
: For over a week, perhaps 3 or 4 weeks Douglas has been plaguing my
: threads with his obnoxious posts. Douglas for some reason thinks he
: knows math, and for several reasons feels that I do not know math. And
: so he has begun to plague my posts with his obnoxious dribble. Like
: telling me I should get off of Internet and learn math from a Community
: College and other assorted innuendos and dart throwing at me.
********************************
A.P.: if you can't stand the heat--get out of the kitchen. You should
know that the issue of who is plaguing whom is relative. You lambast Zare
for posting in your threads at the same time that you dominate my thread
with you frequent post. One could interpret that as hypocrisy.
Your personal feeling toward Zare or anyone else who posts here is your
private business: please leave it that way. I have been very tolerant of
your posts for sometime now. I've often been thought of (unfairly) as a
crack pot myself. Your contrarian views are OK with me--your acrimonious
acts on others are not! So, if you're such a master of math logic, please
prove it by adhering to Aristotelian logic that forbids ad hominem
arguments.
Now, I have a question for you? If I understand you correctly that the
Naturals are ill-defined, then tell me in precise terms why Peano's
postulates of the Naturals don't really work.
cheers
-- patrick <re...@xroads.com>
Archimedes Plutonium
j...@a.cs.okstate.edu (John Chandler) writes:
> Not true at all, of course!
>
> This is a common misconception of beginning students.
>
> (Do you know the first counterexample to this?
You , John Chandler do not belong in math. Perhaps applied math, or
maybe a life insurance actuary. In math proving, look, but stay out.
Archimedes Plutonium
kov...@mcmail2.cis.McMaster.CA (Zdislav V. Kovarik) writes:
> Did everyone notice the shifting of responsibility for definitions and
> proofs form A.P to his/her/its/their opponents? (Not to mention the
> clowning with numbers 94 and 231). A.P. is simply baiting you, and has
> acquired such a mastery in this that he/she/it/they can be awarded the
> degree of Master Baiter.
>
> Have fun, ZVK (Slavek)
I forgive myself. For recently I engaged in a dialogue with ZVK on
Internet. Many of my past posts sept93-may94 and related posts are not
yet all assimilated. That is a file approaching 24MB of storage. That
is my excuse.
I am forbidden to have anything to do with people I tell to go to
hell. I told ZVK to go to hell a long time ago. I forgive myself. But I
know that I rightfully should spend my time on positive things, not on
negatives and their spew, and that is my excuse. It will not happen
again concerning ZVK. So spew all you want ZVK. To tell you the honest
truth, I enjoy those flamers from Acheron and Styx. Hell is a hot
place. And as many who follow my posts can see that once I tell someone
to go to hell, they seem to not bother me anymore or at least less
frequently. This might be because there are some viewers who email
around messages about certain Net characters and odds are there are
persons who email a spiel about me. I can imagine that this email would
say something like, "watch what you post to AP, for it is not known
whether he is a genius and if you post bad then it will probably come
back into your face in the future." I say this because when I first
came to the Internet someone emailed me from Texas saying who Abian
was. I later found out the circumstantial contrary.
Archimedes Plutonium
jgr...@frodo.d.umn.edu (John Greene) writes:
>
> I can think of at least two other possibilities!
Thanks for the post on adics. But this thread strayed into the old
Naturals = foggy finites. It strayed into the Euclid proof of
infinitude of primes with no connection to Adics except for my
insistence that no primes exist in adics.
But thanks for the information on 10-adics for that is true math
knowledge. It is Naturals = Adics = Infinite Integers. John, do you see
a clear cut way of defining Math Induction, Primeness all from
Reals,ie, the Whole Reals such as 2.000.... and 3.00.... etc? Define
primeness from Whole Reals and then carry that definition over to Adics
where 10 adics are 2 Real Whole prime and 5 also. What we lose in this
is almost nothing as far as concepts. But what we gain is PRECISION of
definition. And isn't that what math is all about-- PRECISION. For math
is the science of PRECISION.
Dik T. Winter
> In article <D9s4M...@cwi.nl>
> d...@cwi.nl (Dik T. Winter) writes:
> > Why can I not conclude that given P {the complete list of primes} and
> > M is not in P, that M is not prime?
>
Do you claim the conclusion is false? If so, why? You did not give a
reason why my conclusion was false.
> ... Not until you pick
> up a new prime can you say "contradiction". You can say contradiction
> once you have found a new prime not on that list of All primes. You
> can say contradiction then, because obviously it is not All primes.
Hold a moment. If I have a premissa P and by valid proof from P I can
conclude Q, but also not Q. In that case I can say contradiction.
I need not necessarily contradict the premissa, I need only show that
if the premissa holds I can arrive at two conflicting results.
>
> Say you accept M as not prime because of your earlier steps saying
> that you had the complete list of the primes. What are you going to do
> next Dik? You still need to produce a prime not on that list of all
> primes. Only by producing a new prime not on the list of all primes can
> you ever utter "contradiction".
Nope. You only need to show that the premissa (this is the full list of
primes) is in error. You do it by showing that if you assume the premissa
you can arrive at two conflicting results by valid reasoning.
You do not come up with a new prime, you come up with "a new prime that is
however not in the complete list of primes, under the condition that the
number of primes is finite and completely listed". I come up with a number
that is "not prime but not factorizable, under the condition that the number
of primes is finite and completely listed". (By chance they are the same
number, but that is something else.) Both show that the condition must
be wrong. Your reasoning is: "here I have a complete list, I find a new
number satisfying the properties, but hey: it is not in the list; something
must be wrong". My reasoning is: "here I have a complete list, I find a new
number that should be divisible by one in the list, but hey: it is not
possible, something must be wrong".
> It is not irrelevant. The Fundamental Theorem of Arithmetic (UPF) is
> completely connected with this indirect proof. UPF gives that the two
> new numbers W-1 and W+1 are Necessarily prime.
Under the condition that the number of primes is finite and completely
listed.
> You are not done. You have not produced a new prime.
You have neither. You have only produced one under the above condition
(which you try to proof false).
...
> (BTW, some viewers have emailed me saying I must say something about
> W+1 is not equal to 1.
You probably misunderstood. Most will have talked about "unit", not 1;
at least I did. A unit is a number that has an inverse. In the 10-adics
all numbers except numbers that are multiples of 2 and 5 are units. That
is why in the 10-adics only 2 and 5 are prime; units are not prime. In the
"counting numbers" only 1 is a unit.
Archimedes Plutonium
za...@cco.caltech.edu (Douglas J. Zare) writes:
> You ridiculed Erdos and others. This demonstrates that your ignorance. In
Your crime is that of deifying those; that are just average people.
> fact, one early theorem of Erdos (was it when he was a teenager?) was
> that there exists a prime between n and 2n for every natural n.
Get your facts straight, Chebyshev proved it long before Erdos was even
born. And wise people recognize that the world is full of
idiot-savants. That is, people who have a specialty talent in something
yet never really contribute to the subject per say.
Archimedes Plutonium
> Archimedes Plutonium <Archimedes...@dartmouth.edu> wrote:
> >[...]
> >wrinkle has been written into it since Euclid. Well, I am sorry to say
> >but noone before 1993 could even give a valid proof of it.
>
> There are many, many proofs that there infinitely many primes. You have
Douglas Zare trying to prove Euclid's reductio ad absurdum of
infinitude of primes. First he goes with the flow of Hardy and all the
invalid attempts of counterexample way.
AP: Hardy's A MATHEMATICIANS APOLOGY are wrong and invalid for they
argue
AP: that there is a prime factor that may be missing. I argue in my
proof
AP: that (P1 x P2 x ... x Pk) + 1 is necessarily prime.
DZ: Did you not understand/believe the counterexamples?
And then he realizes he is wrong and so backpedals.
DZ on 5Jun95 wrote:
> The new number is also necessarily not prime. In the natural numbers you
> get a contradiction. The type of contradiction is irrelevant. There are
> many instances of several different proofs being given in popular
> mathematics books; why does this one bother you?
Okay, I will tell you why it bothers me. How much do Caltech students
pay to hear arrogant peacocks like you teaching? You Douglas Zare
cannot even do Euclid's infinitude of primes. Just look at your above.
Can students at Caltech get their $30,000. back?
Archimedes Plutonium
I concur Dik.
Say Dik, if I make the doubly infinites as vectors, wherein the
leftward string is the direction and the rightward string the size
(realizing that the size S is 0<S<1 ) in such a way that each of these
vectors is the arctanh of a Ordered Real/Complex number?
Douglas J. Zare
>threads with his obnoxious posts.
When hundreds of newsservers carry your vile prose, it is not your
thread. You post absurdities to sci.math; I correct them and ask you not
to post again. I believe that your efforts are a deliberate attack on
this newsgroup.
>Douglas for some reason thinks he
>knows math, and for several reasons feels that I do not know math. And
Reality check: Most other readers of sci.math feel the same way. You find
mathematics foggy and wrong. Is it a coincidence or a conspiracy that we
all agree when we disagree with you? Maybe we are studying and understanding
something you are not. You do not seem to try.
>so he has begun to plague my posts with his obnoxious dribble. Like
>telling me I should get off of Internet and learn math from a Community
>College and other assorted innuendos and dart throwing at me.
>[...]
> What is my stereotype of Douglas Zare? It is this. Douglas I envision
[Unimaginative flame deleted.]
>great at computation. But when it comes to MATH LOGIC, they slow down
>to a grinding halt.
In none of my papers have I used anything which would be called computation.
You see mathematics from the perspective of a high school student.
However, most of those who care have some inkling that there is
mathematics beyond popular mathematics books, and there is something
mathematicians take years to study.
> You see Douglas now realizes he is out of his league here. And that
>is a pitiful shame and it goes to show that the Princeton graduate
>students are far, far better than what ever Cal tech seems to offer up
>to me as a math person to expose.
Is that why Princeton accepted me? Do you really believe all of the
ridiculous things you say?
> You see, Douglas is now realizing that he is an embarrassement to
>math, and to math at Caltech. So, please tell us Dougy baby are you
>just a Caltech graduate student or are you a Caltech professor? And
>give us those counterexamples for now you are no longer and
>regurgitation and computation turf. You are on my forte, math logic,
>and, you have no chance of winning against me. I have already made
>mincemeat out of your hobgoblin math illogic. Post your
>counterexamples, DIMWIT Douglas.
I concede I have no hope of proving the things you claim to prove. I do
not know where to start proving that there are only finitely many natural
numbers. Nor can I figure out how to show that 59*509 is prime. My
intuition does not help me to find line-preserving bijections between
S^2, E^2, and H^2. I'll have to ask you for help in showing that all
infinite sets have the same cardinality. I can't find the contradiction
in the Peano Axioms. Though I have studied and used them since I was 14,
I still cannot show that finite fields, with characteristic p, are really
p-adics, with characteristic 0. I still do not understand the
inequivalency of contradictions. In fact, even with the help of all of
the rest of the mathematical community, I cannot find counterexamples to
Fermat's Last Theorem.
However, perhaps my inability to prove patently false statements means
that I may be fit to be a mathematician, but not a crackpot.
Douglas Zare
John Greene
>
> Well, our "INVERSE ASSUMPTION" is that the set of primes is finite. We
> applied addition and multiplication to all the numbers in this set,
> and we calculated another number, X. Our "DERIVED RESULT" is that X
> is not a part of our set of primes, and that X is not evenly divisible
> by any of the numbers in our set of primes. If a number is not
> evenly divisible by any prime (except itself), then it is by definition
> a prime.
>
I've pointed this out before but it is worth repeating: This is NOT the
definition of a prime. A number is prime if it can not be written as
a product of two integers which are both larger than 1. Said differently,
n is prime if whenever n = km where k and m are positive integers, then
k = 1 or m = 1. There is nothing in the definition of a prime number
that says that if a number is not divisible by any prime (except itself)
then it is prime.
This result is true, but it is a derived fact, following from the fundamental
theorem of arithmetic.
John
Earle Jones
To put the math in terms that even Dartmouth students will understand:
CalTech 1
Dartmouth 0
=============
Earle Jones
Earle Jones
Earle Jones
>Archimedes Plutonium (Archimedes...@dartmouth.edu) wrote:
>: In article <3qs5oq$3...@dartvax.dartmouth.edu>
>
>: > In article <3qp9ti$6...@gap.cco.caltech.edu>
>: > za...@cco.caltech.edu (Douglas J. Zare) writes:
>: >
>:
>: For over a week, perhaps 3 or 4 weeks Douglas has been plaguing my
>: threads with his obnoxious posts. Douglas for some reason thinks he
>: College and other assorted innuendos and dart throwing at me.
>
>
>A.P.: if you can't stand the heat--get out of the kitchen. You should
>know that the issue of who is plaguing whom is relative. You lambast Zare
>for posting in your threads at the same time that you dominate my thread
>with you frequent post. One could interpret that as hypocrisy.
>
>Your personal feeling toward Zare or anyone else who posts here is your
>private business: please leave it that way. I have been very tolerant of
>your posts for sometime now. I've often been thought of (unfairly) as a
>crack pot myself. Your contrarian views are OK with me--your acrimonious
>acts on others are not! So, if you're such a master of math logic, please
>prove it by adhering to Aristotelian logic that forbids ad hominem
>arguments.
>
>Now, I have a question for you? If I understand you correctly that the
>Naturals are ill-defined, then tell me in precise terms why Peano's
>postulates of the Naturals don't really work.
>
>cheers
>
>-- patrick <re...@xroads.com>
>
>
John Greene
Archimedes...@dartmouth.edu (Archimedes Plutonium) wrote:
> Thanks for the post on adics. But this thread strayed into the old
> Naturals = foggy finites. It strayed into the Euclid proof of
> infinitude of primes with no connection to Adics except for my
> insistence that no primes exist in adics.
> But thanks for the information on 10-adics for that is true math
> knowledge. It is Naturals = Adics = Infinite Integers. John, do you see
> a clear cut way of defining Math Induction, Primeness all from
> Reals,ie, the Whole Reals such as 2.000.... and 3.00.... etc? Define
> primeness from Whole Reals and then carry that definition over to Adics
> where 10 adics are 2 Real Whole prime and 5 also. What we lose in this
> is almost nothing as far as concepts. But what we gain is PRECISION of
> definition. And isn't that what math is all about-- PRECISION. For math
> is the science of PRECISION.
I'm sure I can not do as you ask, but I can say this: One usually says that
the p-adics have a single prime--p. In general, the n-adics have as many
primes as n has prime factors. Since 10 = (2)(5), the 10-adics have two
primes 2, 5. In order for this to make sense, we must have some notation-
a number is called a unit if its reciprocal is also a number. In the 10-adics,
3 is a unit since 1/3 = ...666666666667. On the other hand, 22 is not a
unit-there is not 10-adic number that you can multiply by 22 to get 1. In
general, it only makes sense to try to factor nonunits. You factor them into
a product of primes times a unit, so 22 = (11)(2). 11 is a unit, 2 is prime.
As another example, 600 = (3)(2)(2)(2)(5)(5). The 2's and 5's are primes,
3 is a unit.
In Euclid's proof that there are infinitely many primes, he implicitely
uses the fact that in the "old naturals," 1 is the only unit. Thus, if
you have a number > 1, it is not a unit, so it factors into primes. This
does not work with the 10-adics since they have infinitely many units.
John
Douglas J. Zare
>j...@a.cs.okstate.edu (John Chandler) writes:
>
>> Not true at all, of course!
>> This is a common misconception of beginning students.
>> (Do you know the first counterexample to this?
>
> You , John Chandler do not belong in math. Perhaps applied math, or
>maybe a life insurance actuary. In math proving, look, but stay out.
Why does the only owner of a glass house in this thread throw stones?
John Chandler's statements seemed mathematically competent to me; they
merely provoked AP's often observed reaction to being confronted with his
errors and vague statements. AP attacked blindly, off-topic, and without
correct punctuation.
(To inject some mathematics) Heuristically, estimate the number of primes
one obtains by taking one more than the nth partial product of the
primes, for n=1 to 10,000.
Douglas Zare
Archimedes Plutonium
jgr...@frodo.d.umn.edu (John Greene) writes:
> I've pointed this out before but it is worth repeating: This is NOT the
> definition of a prime. A number is prime if it can not be written as
> a product of two integers which are both larger than 1. Said differently,
> n is prime if whenever n = km where k and m are positive integers, then
> k = 1 or m = 1. There is nothing in the definition of a prime number
> that says that if a number is not divisible by any prime (except itself)
> then it is prime.
This thread is too long. After today I cease from posting to it. I
understand your definition. But I want to ask a few unrelated questions
surrounding that definition. How about negative integers? And is the
Infinitude of Primes nullified once we consider all integers,
apparently yes. And, even though you are probably not interested John,
but are there counterexamples for FLT when negative integers are
included? I can see 1^3 + (-1)^3 = 0^3, and for all odd exponents. But
what about even exponents such as 4?
I just wonder if the reason the Riem Hyp is such a quagmire
concerning the negatives is because the math community just pretends
for the most time that negatives are ignored? What is the Riem Hyp when
it is converted to Naturals = Adics?
Archimedes Plutonium
ejo...@worm.hooked.net (Earle Jones) writes:
> =============
>
> To put the math in terms that even Dartmouth students will understand:
>
> CalTech 1
> Dartmouth 0
>
> =============
What wind blew in this blow fly named Earle Jones. Not only do you
not understand math, for in that above dialogue with Dik I was
concurring the fact about 10-adics whereas the posts, those posts were
in the old Naturals = Finite Integers where the only unit is 1. That is
what I was concurring with Dik. And I no longer wanted to argue with
Dik over the Euclid IP for I respect Dik too much to care to continue
over IP.
But you Earle Jones are trying to propagandize a smear campaign
against me. And I simply do not have the time to defend myself against
jerks like you. I will be quick to tell you to go to hell if you keep
up this bent.
I am no longer going to post to this thread for it is already too
long and so I will continue the discussion of Euclid Infinitude of
Primes in new separate threads.
Stop your smear campaign, and try to learn some math. Forget about
who you like and dislike, for math is impersonal, learn some math.
Archimedes Plutonium
ejo...@worm.hooked.net (Earle Jones) writes:
> =============
>
> To put the math in terms that even Dartmouth students will understand:
>
> CalTech 1
> Dartmouth 0
>
> =============
This is the fourth post with your horsemanure in it. I bet you don't
even know a Euclid proof of IP. And yet somehow out of the tone of the
posts you have drawn some idiotic conclusion.
Here is Zare's proof of Euclid's Infinitude of Primes. And it is not
even a proof. But that is the best that Zare can do.
DZ: Did you not understand/believe the counterexamples?
And then he realizes he is wrong and so backpedals.
DZ on 5Jun95 wrote:
> The new number is also necessarily not prime. In the natural numbers you
> get a contradiction. The type of contradiction is irrelevant. There are
> many instances of several different proofs being given in popular
> mathematics books; why does this one bother you?
The next time that you Earle Jones smear campaigns falsely against
me, I will tell you to go to hell for I just simply do not have the
time for straighten-out hordes of nitwits who do not know the
difference between their head and shinola.
Archimedes Plutonium
za...@cco.caltech.edu (Douglas J. Zare) writes:
> When hundreds of newsservers carry your vile prose, it is not your
> thread. You post absurdities to sci.math; I correct them and ask you not
> to post again. I believe that your efforts are a deliberate attack on
> this newsgroup.
You can not even deliver a Euclid Infinitude of Primes and that is why
I think you are a math fraud.
You implied that you knew the conversions from Loba to Eucl geom and
vice versa, and come to find out you did not know them.
In my opinion you Douglas Zare are a math fraud and in my opinion a
liar.
If you do not like my posts, just ignore them. Or put me your killfile.
If you continue to pester me such as the email you keep sending to my
sysadm as shown below, I am going to get to the point where I will tell
you to go to hell. I just simply do not have the time to be pestered.
I will no longer post to this thread, for it is too long.
And do not send me or my sysadm anymore of your dumb email.
EMAIL
From: za...@cco.caltech.edu (Douglas J. Zare)
Date: Fri, 9 Jun 1995 03:18:39 -0700
To: Archimedes...@Dartmouth.EDU
Subject: Re: DIFTA,AFTA,crab apple blossoms,perennial bachelor buttons,
..
Newsgroups: alt.sci.physics.plutonium,sci.math
In-Reply-To: <3r7nvn$j...@dartvax.dartmouth.edu>
Organization: California Institute of Technology, Pasadena
Cc: ro...@dartvax.dartmouth.edu, za...@cco.caltech.edu
AP, your post had nothing to do with mathematics. You are free to
worship
as you please, but this is clearly inappropriate for sci.math. Do you
have any idea how much resources you waste? How many people read
sci.math
for songs about your religion?
Please encourage AP to have some responsibility for his posts. His
attack
on sci.math continues with several bizarre and nonmathematical posts
each
day. Please tell me if you would prefer me to take action in another
manner.
Thank you for your attention.
Douglas Zare
za...@cco.caltech.edu
In article <3r7nvn$j...@dartvax.dartmouth.edu> you write:
> Rejoice, it is spring and the work of finding Doubly Infinites =
>points of Loba geom and the sight of glorious crab apples in blossoms;
>red snap dragons; the bluish argeratum; hills of forget-me-nots; and
>the find of AFTA once DIFTA is got where Adics = points of Riem geom;
Archimedes Plutonium
za...@cco.caltech.edu (Douglas J. Zare) writes:
> Why does the only owner of a glass house in this thread throw stones?
> John Chandler's statements seemed mathematically competent to me; they
> merely provoked AP's often observed reaction to being confronted with his
> errors and vague statements. AP attacked blindly, off-topic, and without
> correct punctuation.
My last post to this thread. It is far too long. Let it die.
True, I sometimes go overboard. And thanks DZ for kinda soothing John
Chandler and making him feel better as though he belongs in math. But
not to the point that he takes your place there at caltech, financing
and all and Zare goes to okstate.
I could have told John Chandler more politely instead of my crushing
style. I should have overlooked John's indignations and baby talk type
of post. I should not haves said in two lines that he had it all
screwed-up. I could have said, Please reconsider what it means to
reductio ad absurdum. I had watched that B movie with Robin Williams
where a radio announcer had caused berserkness. And I too am aware
that if I crush some math weakling it may send him/her over the brink.
The Internet may do that to some person. So we must be on guard for
that.
I could have shown John Chandler Ron Bruck's post of 1993, as shown
below. Because I would say that 95% of the people involved in math get
their math understanding not from their thinking but from what other
people say, rather than from their own mental thoughts. For if Paul
Erdos were to come onto the Internet and say that Hardy's IP was flawed
and AP's IP was valid, then overnight the situation would be changed
and all the flawed IP's written in textbooks would then start to
disappear.
Here all of you math haters, heed the words of Ron Bruck. For Ron is
after the math understanding.
From: br...@mtha.usc.edu (Ronald Bruck)
Newsgroups: sci.math
Subject: Re: PROOF OF THE INFINITUDE OF
Date: 27 Aug 1993 21:59:06 -0700
Organization: University of Southern California, Los Angeles, CA
Lines: 52
Message-ID: <25momq$2...@mtha.usc.edu>
>In article <25m09f$3...@umd5.umd.edu> sto...@oyster.smcm.edu (Stanley Toney) writes:
>>I think that your IP proof shows that if primes are assumed
>>finite then with the product of {2,3,5,7. . .LP}=W (LP being last >>prime) W+1 and or W-1 is prime. but I don't think it shows that >>both must be prime.
>Not even that much is true. W=2x3x5x7x11x13x17 has both W-1
>and W+1 composite.
I am really getting fed up with the incredible number of posts
which keep making this assertion. There is a HYPOTHESIS: THAT
THE NUMBER OF PRIMES IS FINITE. From that there is a DEDUCTION:
THAT W-1 AND W+1 ARE PRIME. The reasoning is CORRECT. You cannot refute
the reasoning by exhibiting an example where W-1 and W+1 are both
composite; that is not the point; the point is that an IMPLICATION is
being asserted. You cannot refute it by presenting alternative
reasoning which shows that W-1 and W+1 are composite; that is not the
point; it does not address the implication which is being asserted. It
is irrelevant that your alternative reasoning reaches a conclusion
contradicting LP's conclusion; OF COURSE there are contradictions;
THAT'S THE POINT of a reductio ad absurdum argument.
My God, this is sci.math. People here are supposed to be able to
distinguish between the proposition A==>B and the proposition B. Did
none of these people take a logic course? A logic course shouldn't be
necessary! To be a mathematician is to have this ingrained into your
very bones from birth! ! That wasn't dyslexia when you wrote the
backward E, you were reinventing the existential quantifier!
John Greene
Archimedes...@dartmouth.edu (Archimedes Plutonium) wrote:
> This thread is too long. After today I cease from posting to it. I
> understand your definition. But I want to ask a few unrelated questions
> surrounding that definition. How about negative integers? And is the
> Infinitude of Primes nullified once we consider all integers,
> apparently yes. And, even though you are probably not interested John,
> but are there counterexamples for FLT when negative integers are
> included? I can see 1^3 + (-1)^3 = 0^3, and for all odd exponents. But
> what about even exponents such as 4?
> I just wonder if the reason the Riem Hyp is such a quagmire
> concerning the negatives is because the math community just pretends
> for the most time that negatives are ignored? What is the Riem Hyp when
> it is converted to Naturals = Adics?
>
Concerning negative integers, I'm not sure what your question is, but I
will take my best shot. If one wishes to discuss primes and factorization
within the set of all "old fashion" integers rather than just the positive
ones, then there are two units: 1, -1. The definition is rewritten as
p is prime if whenever p = mn, either m or n is a unit. For example, we
say that -7 is prime even though it has the factorization -7 = (-1)(7),
because -1 is a unit. We call -7 and 7 associates. In general, two
numbers are called associates if their quotient is a unit. Usually, when
you list primes, you do not list associates, so the list of primes is still
taken to be {2, 3, 5, 7, 11, ...} but it is understood that -2, -3, -5, ...
really are primes also.
For the 10-adics, there are actually infinitely many primes. Examples are
2, 6, 14, 22, 102, etc. But all primes are associates to either 2 (which was
the case for the examples I gave) or 5. If you only list the nonassociate
primes, then the list for the 10-adics is {2, 5}.
If 0 is allowed in Fermat's last theorem, then 1^n + 0^n = 1^n, or even
0^n + 0^n = 0^n works for all positive integers n. Just as the definition
of prime is changed slightly if one talks about sets other than the positive
integers, so to the statement of Fermat's last theorem is changed. Here is
one formulation: If a^n + b^n = c^n, where a, b, c, n are integers and n > 2,
then abc = 0.
If one restricts n to be an odd prime (usual) then this is usually written:
The only solutions to a^p + b^p + c^p = 0 are those in which one of a, b, c
is 0.
I can't answer your questions on the Riemann Hypothesis.
John
Archimedes Plutonium
Denis Pihan <dmp> writes:
> An open invitation to Archimedes.Plutoninium to extend his 'theorem'
> that 'finite numbers = adics' to the theory of polynomials:
>
> You claim that no one in the mathematical community has ever
> given a definition of finite (other than your esteemed self).
> You have on a number of occasions made the claim that
> 'finite numbers = adics'.
>
> Do you similarly claim that there is no definition of 'finite
> polynomial' - i.e. polynomial of finite degree? Would you
> therefore care to prove 'finite polynomials = adic polynomials'
> (= formal power series),
> i.e. polynomials of the form . . . + a_n x^n + . . . + a_1 x + a_0?
> If not, what constitutes a polynomial of 'finite' degree? What
> kind of 'finite' number is implied in this definition
My claim is that the old thinking of Naturals = Finite Integers is
garbage. It never had a definition of "finite". And it will be replaced
by the true Naturals where Naturals = Adics = Infinite Integers. The
old way was as silly as thinking that the set of Reals are only those
Reals which start repeating infinitely rightwards with zeros. Thus 3 is
a Real and 3.5 is a Real but 3.33... or pi are not Reals. Try building
a math on these silly Reals.
The best way to prove this claim of mine is my present project. I am
about to show that each adic number from all adics are the intrinsic
geometrical points of a Riemannian sphere. And that the Doubly
Infinites, unknown as to what they are as of 1993, are the actual
instrinsic geometrical points of Lobachevskian geometry. We already
know that the Complex plane is Euclidean geometry. Once I sort this
mess out. You will see that what separates Adics from Doubly Infinites
is imaginary radius; what separates Adics from Eucl is tan and what
separates Doubly Infinites from Eucl is tanh. This is all very
beautiful work and ideas.
One of the results of this beautiful math will be the realization
that a number exists if it is the point of a geometry. If it is not the
point, the actual point of a geometry then the number is just a made-up
illusion, a Loch Ness monster number. When this project is done, then
one of its implications is that there exist 3 and only 3 separate and
distinct types of numbers with nothing in-between. So then the Naturals
= Finite Integers was just a chimera all along. What remains is that
Whole Reals of 1.000.... , 2.0000...., 3.00... as a subset replaces our
old goofballish idea of Naturals. And it is apparent that Reals have a
finite string leftward and theorems such as FLT in Whole Reals are
ill-defined.
The adics are already polynomials and are all of them!!
> Could you please provide the mathematical definition of 'algebraic'
> and 'transcendental' consistent with these 'new' definitions of
> polynomials, or alternatively explain why
> finite polynomials <> adic polynomials?
This is difficult for it entails a new definition for the idempotents
in 10-adics and the i in 5-adics are obviously simultaneously
transcendental and algebraic. As one poster to sci.math posted upon
seeing the 5-adic sqrt of -1 asking what kind of number it is. So then
new definitions must be mustered forth. This is as it should be when a
math revolution is started, much of the old is seen as decayed and
inadequate and be ready to be trashcanned for the new.
David Kastrup
:An open invitation to Archimedes.Plutoninium to extend his 'theorem'
:that 'finite numbers = adics' to the theory of polynomials:
:You claim that no one in the mathematical community has ever
:given a definition of finite (other than your esteemed self).
:You have on a number of occasions made the claim that
:'finite numbers = adics'.
:Do you similarly claim that there is no definition of 'finite
:polynomial' - i.e. polynomial of finite degree? Would you
:therefore care to prove 'finite polynomials = adic polynomials'
:(= formal power series),
... rest deleted.
What sense is there in this request? LP is happy with mixing up
definitions and not hearing to the advice of anybody to not switch
axioms and demand lasting validity in scalars already. What sense is
there in "challenging" him into more difficult waters, when he is
eager to swallow heavily in pools already?
--
David Kastrup, Goethestr. 20, D-52064 Aachen Tel: +49-241-72419
Email: d...@pool.informatik.rwth-aachen.de Fax: +49-241-79502
Danny Calegari
inexplicably attached as the author of this article:
An open invitation to Archimedes.Plutoninium to extend his 'theorem'
that 'finite numbers = adics' to the theory of polynomials:
You claim that no one in the mathematical community has ever
given a definition of finite (other than your esteemed self).
You have on a number of occasions made the claim that
'finite numbers = adics'.
Do you similarly claim that there is no definition of 'finite
polynomial' - i.e. polynomial of finite degree? Would you
therefore care to prove 'finite polynomials = adic polynomials'
(= formal power series),
i.e. polynomials of the form . . . + a_n x^n + . . . + a_1 x + a_0?
If not, what constitutes a polynomial of 'finite' degree? What
kind of 'finite' number is implied in this definition?
You have claimed in the past that there is a distinction between
algebraic and trancendental numbers, and you have claimed for
instance that e and pi are transcendental. However, pi is a 'root'
of the formal power series x - x^3/3! + x^5/5! - x^7/7! . . .
and e is a 'root' of the power series determined by the expansion
(x-1) - (x-1)^2/2 + (x-1)^3/3 - (x-1)^4/4 + . . . which can
be rearranged easily to give a formal power series? (Note: this
power series in practice does not converge for x=e, but it can
be extended to an analytic function in C which has a zero at x=e)
Could you please provide the mathematical definition of 'algebraic'
and 'transcendental' consistent with these 'new' definitions of
polynomials, or alternatively explain why
finite polynomials <> adic polynomials?
Regards,
Danny Calegari
Archimedes Plutonium
Danny Calegari <dan...@mundoe.maths.mu.oz.au> writes:
Are there finite polynomials? No, because the infinite 0's of unlisted
terms are just conveniently lopped-off and unmentioned.
Are there finite matrices? No, because the infinite 0's are
conveniently removed.
Are there finite numbers? No. We have the Reals. They are all infinite.
There seems to be no way in which to define "finite" without infinite
place value.
All Reals are infinite expansions.
All Adics are infinite expansions.
Sorry I can not answer anymore on this topic. I am working on Adics =
the actual geometrical points of Riem geom and Doubly Infinites = the
actual geometrical points of Loba geom. Once I have that, noone will
never need a proof of any of the above Danny because "Numbers exist if
and only if they are infinite and if and only if they are a actual
point of one of the 3 and only 3 geometries." That work will be the
greatest math work to date for it will be obvious that Number exists if
and only if it is a point of either Eucl, or Riem or Loba geom. No
finite number makes any sense when that work is given. That work will
make the old Naturals, the Naturals = Finite Integers as primitive to
math as primitive as witchdoctoring is to modern medicine.
Sorry Danny but I just do not have the time to be diverted astray.
But thanks anyway.
Kin Yan Chung
John Greene <jgr...@frodo.d.umn.edu> wrote:
>If 0 is allowed in Fermat's last theorem, then 1^n + 0^n = 1^n, or even
>0^n + 0^n = 0^n works for all positive integers n. Just as the definition
>of prime is changed slightly if one talks about sets other than the positive
>integers, so to the statement of Fermat's last theorem is changed. Here is
>one formulation: If a^n + b^n = c^n, where a, b, c, n are integers and n > 2,
>then abc = 0.
I understand that it is this formulation of FLT that Wiles proved. In this
case, Plutonium's "counterexamples" aren't even 10-adic counterexamples:
when a and b are the two distinct idempotents of the 10-adic numbers, then
ab=0. Under the ludicrous assertion that "naturals = adics" (whatever that
means), Plutonium's offering does not contradict the above formulation of
FLT at all, and more specifically it does not contradict Wiles' result.
--
Kin Yan Chung (kin...@math.princeton.edu) | Sydney _--_|\
Math Department, Princeton University, Princeton NJ 08544 | 2000 / \
WWW Home Page: http://www.princeton.edu/~kinchung | \_.--._/
I've got this vi thing worked out.... :w :q :wq! ZZ ^Z ^D ^[[1 ^H ^C^C v
Douglas J. Zare
>John Greene <jgr...@frodo.d.umn.edu> wrote:
>>[...]
>>one formulation: If a^n + b^n = c^n, where a, b, c, n are integers
>>and n > 2, then abc = 0.
>
>I understand that it is this formulation of FLT that Wiles proved. In this
>case, Plutonium's "counterexamples" aren't even 10-adic counterexamples:
It might be worth saying that for most n, it is not rare for a p-adic
(hence 10-adic) to have an nth root, whether it is the sum of two nth
powers or not.
Douglas Zare