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First Proof That Infinitely Many Prime Numbers Come in Pairs
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Sam Wormley
May 16, 2013, 12:03:24 PM5/16/13
to
First Proof That Infinitely Many Prime Numbers Come in Pairs
> http://www.scientificamerican.com/article.cfm?id=first-proof-that-infinite-many-prime-numbers-come-in-pairs
> That goal is the proof to a conjecture concerning prime numbers.
> Those are the whole numbers that are divisible only by one and
> themselves. Primes abound among smaller numbers, but they become less
> and less frequent as one goes towards larger numbers. In fact, the
> gap between each prime and the next becomes larger and larger — on
> average. But exceptions exist: the ‘twin primes’, which are pairs of
> prime numbers that differ in value by 2. Examples of known twin
> primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2^195,000 − 1 and
> 2,003,663,613 × 2^195,000 + 1.
>
> The twin prime conjecture says that there is an infinite number of
> such twin pairs. Some attribute the conjecture to the Greek
> mathematician Euclid of Alexandria, which would make it one of the
> oldest open problems in mathematics.
> http://www.scientificamerican.com/article.cfm?id=first-proof-that-infinite-many-prime-numbers-come-in-pairs
> That goal is the proof to a conjecture concerning prime numbers.
> Those are the whole numbers that are divisible only by one and
> themselves. Primes abound among smaller numbers, but they become less
> and less frequent as one goes towards larger numbers. In fact, the
> gap between each prime and the next becomes larger and larger — on
> average. But exceptions exist: the ‘twin primes’, which are pairs of
> prime numbers that differ in value by 2. Examples of known twin
> primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2^195,000 − 1 and
> 2,003,663,613 × 2^195,000 + 1.
>
> The twin prime conjecture says that there is an infinite number of
> such twin pairs. Some attribute the conjecture to the Greek
> mathematician Euclid of Alexandria, which would make it one of the
> oldest open problems in mathematics.
Pubkeybreaker
May 16, 2013, 7:31:20 PM5/16/13
to
On May 16, 12:03 pm, Sam Wormley <sworml...@gmail.com> wrote:
> First Proof That Infinitely Many Prime Numbers Come in Pairs
>
>
>
> >http://www.scientificamerican.com/article.cfm?id=first-proof-that-inf...
> First Proof That Infinitely Many Prime Numbers Come in Pairs
>
>
>
> > That goal is the proof to a conjecture concerning prime numbers.
> > Those are the whole numbers that are divisible only by one and
> > themselves. Primes abound among smaller numbers, but they become less
> > and less frequent as one goes towards larger numbers. In fact, the
> > gap between each prime and the next becomes larger and larger -- on
> > Those are the whole numbers that are divisible only by one and
> > themselves. Primes abound among smaller numbers, but they become less
> > and less frequent as one goes towards larger numbers. In fact, the
> > average. But exceptions exist: the 'twin primes', which are pairs of
> > prime numbers that differ in value by 2. Examples of known twin
> > primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2^195,000 - 1 and
> > prime numbers that differ in value by 2. Examples of known twin
> > 2,003,663,613 × 2^195,000 + 1.
>
> > The twin prime conjecture says that there is an infinite number of
> > such twin pairs. Some attribute the conjecture to the Greek
> > mathematician Euclid of Alexandria, which would make it one of the
> > oldest open problems in mathematics.- Hide quoted text -
>
> > The twin prime conjecture says that there is an infinite number of
> > such twin pairs. Some attribute the conjecture to the Greek
> > mathematician Euclid of Alexandria, which would make it one of the
>
> - Show quoted text -
This is a gross misstatement of the proof. It did NOT prove that there
were infinitely many prime pairs. What it did prove was that the gap
between primes is FINITELY BOUNDED infinitely often. The bound is 70
x 10^6.
While this will probably be improved it is a long way to proving a
bound of
2.
Wally W.
May 16, 2013, 9:19:22 PM5/16/13
to
On Thu, 16 May 2013 16:31:20 -0700 (PDT), Pubkeybreaker wrote:
>On May 16, 12:03 pm, Sam Wormley <sworml...@gmail.com> wrote:
>> First Proof That Infinitely Many Prime Numbers Come in Pairs
>>
>>
>>
>> >http://www.scientificamerican.com/article.cfm?id=first-proof-that-inf...
>> > That goal is the proof to a conjecture concerning prime numbers.
>> > Those are the whole numbers that are divisible only by one and
>> > themselves. Primes abound among smaller numbers, but they become less
>> > and less frequent as one goes towards larger numbers. In fact, the
>> > gap between each prime and the next becomes larger and larger -- on
>> > average. But exceptions exist: the 'twin primes', which are pairs of
>> > prime numbers that differ in value by 2. Examples of known twin
>> > primes are 3 and 5, or 17 and 19, or 2,003,663,613 � 2^195,000 - 1 and
>> > 2,003,663,613 � 2^195,000 + 1.
>>
>> > The twin prime conjecture says that there is an infinite number of
>> > such twin pairs. Some attribute the conjecture to the Greek
>> > mathematician Euclid of Alexandria, which would make it one of the
>> > oldest open problems in mathematics.- Hide quoted text -
>>
>> - Show quoted text -
>
>This is a gross misstatement of the proof.
A gross misstatement by Sam? Really?!
>On May 16, 12:03 pm, Sam Wormley <sworml...@gmail.com> wrote:
>> First Proof That Infinitely Many Prime Numbers Come in Pairs
>>
>>
>>
>> >http://www.scientificamerican.com/article.cfm?id=first-proof-that-inf...
>> > That goal is the proof to a conjecture concerning prime numbers.
>> > Those are the whole numbers that are divisible only by one and
>> > themselves. Primes abound among smaller numbers, but they become less
>> > and less frequent as one goes towards larger numbers. In fact, the
>> > gap between each prime and the next becomes larger and larger -- on
>> > average. But exceptions exist: the 'twin primes', which are pairs of
>> > prime numbers that differ in value by 2. Examples of known twin
>> > primes are 3 and 5, or 17 and 19, or 2,003,663,613 � 2^195,000 - 1 and
>> > 2,003,663,613 � 2^195,000 + 1.
>>
>> > The twin prime conjecture says that there is an infinite number of
>> > such twin pairs. Some attribute the conjecture to the Greek
>> > mathematician Euclid of Alexandria, which would make it one of the
>> > oldest open problems in mathematics.- Hide quoted text -
>>
>> - Show quoted text -
>
>This is a gross misstatement of the proof.
Oh ... I guess not.
He didn't actually write what he posted.
Maybe he'll want to take credit for vetting it ... or not.
Richard Tobin
May 17, 2013, 6:42:33 AM5/17/13
to
In article <4c7758a5-2b88-4756...@d6g2000yqi.googlegroups.com>,
but it appears to be using "prime pairs" to mean "successive primes",
and "twin primes" to mean "prime pairs where the difference is 2".
That makes the headline misleading because we already knew there were
infinitely many pairs of successive primes.
-- Richard
Pubkeybreaker <pubkey...@aol.com> wrote:
>This is a gross misstatement of the proof. It did NOT prove that there
>were infinitely many prime pairs. What it did prove was that the gap
>between primes is FINITELY BOUNDED infinitely often. The bound is 70
>x 10^6.
I agree that the article (quoted from Scientific American) is unclear,
>This is a gross misstatement of the proof. It did NOT prove that there
>were infinitely many prime pairs. What it did prove was that the gap
>between primes is FINITELY BOUNDED infinitely often. The bound is 70
>x 10^6.
but it appears to be using "prime pairs" to mean "successive primes",
and "twin primes" to mean "prime pairs where the difference is 2".
That makes the headline misleading because we already knew there were
infinitely many pairs of successive primes.
-- Richard
Pubkeybreaker
May 17, 2013, 9:34:46 AM5/17/13
to
On May 17, 6:42 am, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <4c7758a5-2b88-4756-91f1-d59d43bcd...@d6g2000yqi.googlegroups.com>,
> In article <4c7758a5-2b88-4756-91f1-d59d43bcd...@d6g2000yqi.googlegroups.com>,
>
> Pubkeybreaker <pubkeybrea...@aol.com> wrote:
> >This is a gross misstatement of the proof. It did NOT prove that there
> >were infinitely many prime pairs. What it did prove was that the gap
> >between primes is FINITELY BOUNDED infinitely often. The bound is 70
> >x 10^6.
>
> I agree that the article (quoted from Scientific American) is unclear,
> but it appears to be using "prime pairs" to mean "successive primes",
> and "twin primes" to mean "prime pairs where the difference is 2".
>
> That makes the headline misleading because we already knew there were
> infinitely many pairs of successive primes.
Not with a finitely bounded gap between them we didn't.
> Pubkeybreaker <pubkeybrea...@aol.com> wrote:
> >This is a gross misstatement of the proof. It did NOT prove that there
> >were infinitely many prime pairs. What it did prove was that the gap
> >between primes is FINITELY BOUNDED infinitely often. The bound is 70
> >x 10^6.
>
> I agree that the article (quoted from Scientific American) is unclear,
> but it appears to be using "prime pairs" to mean "successive primes",
> and "twin primes" to mean "prime pairs where the difference is 2".
>
> That makes the headline misleading because we already knew there were
> infinitely many pairs of successive primes.
Richard Tobin
May 17, 2013, 10:42:21 AM5/17/13
to
In article <0545f14d-ed7c-48a1...@k6g2000yqh.googlegroups.com>,
-- Richard
Pubkeybreaker <pubkey...@aol.com> wrote:
>> That makes the headline misleading because we already knew there were
>> infinitely many pairs of successive primes.
>Not with a finitely bounded gap between them we didn't.
Yes, that's why the headline is wrong - it doesn't say that.
>> That makes the headline misleading because we already knew there were
>> infinitely many pairs of successive primes.
>Not with a finitely bounded gap between them we didn't.
-- Richard
dull...@sprynet.com
May 17, 2013, 11:54:23 AM5/17/13
to
On Thu, 16 May 2013 16:31:20 -0700 (PDT), Pubkeybreaker
the difficulty of factoring large primes.
Makes you wonder how accurate they are on topics where you
can't immediately see the errors...
Aatu Koskensilta
May 17, 2013, 12:45:37 PM5/17/13
to
dull...@sprynet.com writes:
> Some years ago there was something in Scientific American about
> the difficulty of factoring large primes.
I once read it somewhere that factoring large primes is what is known
> Some years ago there was something in Scientific American about
> the difficulty of factoring large primes.
as an NP ("Non-Polynomial") complete problem.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Gus Gassmann
May 17, 2013, 2:00:05 PM5/17/13
to
On 17/05/2013 1:45 PM, Aatu Koskensilta wrote:
> dull...@sprynet.com writes:
>
>> Some years ago there was something in Scientific American about
>> the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
... which also settles that other notorious problem, proving that
> dull...@sprynet.com writes:
>
>> Some years ago there was something in Scientific American about
>> the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
P = NP. Very impressive.
1treePetrifiedForestLane
May 17, 2013, 8:17:08 PM5/17/13
to
bounded?... does that mean that there are no primes,
further than taht, apart?... seems, I don't know, but
I prefer the twin primes, viz
Brun's constant is not transcendental.
further than taht, apart?... seems, I don't know, but
I prefer the twin primes, viz
Brun's constant is not transcendental.
Bill Taylor
May 18, 2013, 1:41:15 AM5/18/13
to
On May 18, 4:45 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
-- Bantering Bill
> dullr...@sprynet.com writes:
> > Some years ago there was something in Scientific American about
> > the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
It's also at the very top of the oracle hierarchy!
> > Some years ago there was something in Scientific American about
> > the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
-- Bantering Bill
Richard Tobin
May 18, 2013, 9:06:55 AM5/18/13
to
In article <bc48b311-68c0-44cd...@fq2g2000pbb.googlegroups.com>,
are always more the 70 million apart.
It's easy to see that there are arbitrarily large gaps between some
consecutive primes. Consider N factorial: the N-1 integers N!-N ... N!-2
are all composite.
-- Richard
1treePetrifiedForestLane <Spac...@hotmail.com> wrote:
>bounded?... does that mean that there are no primes,
>further than taht, apart?...
No, it means that there isn't a point after which consecutive primes
>bounded?... does that mean that there are no primes,
>further than taht, apart?...
are always more the 70 million apart.
It's easy to see that there are arbitrarily large gaps between some
consecutive primes. Consider N factorial: the N-1 integers N!-N ... N!-2
are all composite.
-- Richard
Graham Cooper
May 18, 2013, 4:48:19 PM5/18/13
to
On May 18, 4:00 am, Gus Gassmann <no...@nospam.com> wrote:
> On 17/05/2013 1:45 PM, Aatu Koskensilta wrote:
>
> On 17/05/2013 1:45 PM, Aatu Koskensilta wrote:
>
> > dullr...@sprynet.com writes:
>
> >> Some years ago there was something in Scientific American about
> >> the difficulty of factoring large primes.
>
> > I once read it somewhere that factoring large primes is what is known
> > as an NP ("Non-Polynomial") complete problem.
>
> ... which also settles that other notorious problem, proving that
> P = NP. Very impressive.
>
No Oracles allowed!
>
> >> Some years ago there was something in Scientific American about
> >> the difficulty of factoring large primes.
>
> > I once read it somewhere that factoring large primes is what is known
> > as an NP ("Non-Polynomial") complete problem.
>
> ... which also settles that other notorious problem, proving that
> P = NP. Very impressive.
>
Herc
--
IS DOG+1 a NUMBER ?
http://blockprolog.com/nat-s-dog.png
1treePetrifiedForestLane
May 18, 2013, 5:36:47 PM5/18/13
to
ah, so, there could still be infinity of twin-primes.
Graham Cooper
May 18, 2013, 6:18:01 PM5/18/13
to
On May 19, 7:36 am, 1treePetrifiedForestLane <Space...@hotmail.com>
wrote:
N!-N
---- = (N-1)! - 1
N
^_^
wrote:
---- = (N-1)! - 1
N
^_^
Richard Tobin
May 19, 2013, 6:16:29 AM5/19/13
to
In article <1a2603a5-a24b-4b98...@k8g2000pbf.googlegroups.com>,
Very good, but what is your point?
-- Richard
Very good, but what is your point?
-- Richard
Charlie-Boo
May 19, 2013, 12:40:22 PM5/19/13
to
Charlie-Boo
May 19, 2013, 12:46:11 PM5/19/13
to
On May 18, 4:48 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
Let Q be (all A)(exists B) LT(A,B) ^ P(B) ^ P(f(B))
Could Q be true and unprovable? Could Q be false and unrefutable?
For which P and f [specific or general] is Q one or the other of these
conditions?
C-B
Charlie-Boo
May 19, 2013, 12:58:00 PM5/19/13
to
heck with silly ZF, and we aren't questioning sets anyway, say we are
using PA.
Then if Q = = PA is consistent the 1st condition holds (for us.)
Now what can be proven in PA? This question is really sad, because
apparently people don't know - does anyone? I say this because when
talking about when certain theorems known to be true in PA apply, they
always say "If it has Peano's axioms" or some dinky "You can express
arithmetic in it." PA means a+b=c = = |- a+b=c , a*b=c = = |- a*b=c
and |-(allx)TRUE(x) where TRUE(a) = = a is a natural number. ( TRUE
means everything as in {x|TRUE} not like a "true sentence".)
In general, if Q is a Godel sentence of PA. In other words . . . etc.
C-B
Graham Cooper
May 19, 2013, 4:31:08 PM5/19/13
to
My Defn:
if THM is an axiom then THM is provable
if A is provable and B is provable
and A^B->THM
then THM is provable
Charlie-Boo
May 19, 2013, 11:15:21 PM5/19/13
to
On May 19, 4:31 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
> On May 20, 2:46 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On May 18, 4:48 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > On May 18, 4:00 am, Gus Gassmann <no...@nospam.com> wrote:
>
> > > > On 17/05/2013 1:45 PM, Aatu Koskensilta wrote:
>
> > > > > dullr...@sprynet.com writes:
>
> > > > >> Some years ago there was something in Scientific American about
> > > > >> the difficulty of factoring large primes.
>
> > > > > I once read it somewhere that factoring large primes is what is known
> > > > > as an NP ("Non-Polynomial") complete problem.
>
> > > > ... which also settles that other notorious problem, proving that
> > > > P = NP. Very impressive.
>
> > > No Oracles allowed!
>
> > > Herc
>
> > > --
>
> > > IS DOG+1 a NUMBER ?
>
> > >http://blockprolog.com/nat-s-dog.png
>
> > Let P be a recursive set and f(I) be a recursive function.
>
> > Let Q be (all A)(exists B) LT(A,B) ^ P(B) ^ P(f(B))
>
> > Could Q be true and unprovable? Could Q be false and unrefutable?
>
> > For which P and f [specific or general] is Q one or the other of these
> > conditions?
>
> > C-B
>
> Define provable!
Pay 'tension! I said "say we are using PA. . . . Now what can be
> On May 20, 2:46 am, Charlie-Boo <shymath...@gmail.com> wrote:
>
>
>
>
>
>
>
>
>
> > On May 18, 4:48 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > > On May 18, 4:00 am, Gus Gassmann <no...@nospam.com> wrote:
>
> > > > On 17/05/2013 1:45 PM, Aatu Koskensilta wrote:
>
> > > > > dullr...@sprynet.com writes:
>
> > > > >> Some years ago there was something in Scientific American about
> > > > >> the difficulty of factoring large primes.
>
> > > > > I once read it somewhere that factoring large primes is what is known
> > > > > as an NP ("Non-Polynomial") complete problem.
>
> > > > ... which also settles that other notorious problem, proving that
> > > > P = NP. Very impressive.
>
> > > No Oracles allowed!
>
> > > Herc
>
> > > --
>
> > > IS DOG+1 a NUMBER ?
>
> > >http://blockprolog.com/nat-s-dog.png
>
> > Let P be a recursive set and f(I) be a recursive function.
>
> > Let Q be (all A)(exists B) LT(A,B) ^ P(B) ^ P(f(B))
>
> > Could Q be true and unprovable? Could Q be false and unrefutable?
>
> > For which P and f [specific or general] is Q one or the other of these
> > conditions?
>
> > C-B
>
> Define provable!
proven in PA? This question is really sad, . .."
This may make a constructive proof actually useful (Martin Davis
demanded that I make my axiomatized proofs of incompleteness in logic
constructive despite the loss from ignoring nonconstructive proofs,
much to my chagrin.)
If we can construct Godel sentences (as opposed to only proving they
exist), then as a bonus we can see if they are equivalent to our Q
Now what's the formula for constructing a Godel sentence? And don't
forget that while Godel thought of only 1, there are 3 in the seminal
"18 Word Proof of the Incompleteness Theorems of Godel, Rosser and
Smullyan".
C-B
Charlie-Boo
May 19, 2013, 11:30:30 PM5/19/13
to
On May 18, 4:48 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
http://forums.cannabisculture.com/forums/ubbthreads.php?ubb=showflat&Number=576091
> http://blockprolog.com/nat-s-dog.png
Graham Cooper
May 20, 2013, 1:28:48 AM5/20/13
to
A0 based on the defn I gave you.
!a0(a1)
8203215
Note a 2nd line is required A1=8203215 (in binary)
A FULLY CODED GODEL STATEMENT
https://groups.google.com/group/alt.bible/browse_thread/thread/4f4dc0f290a90d12/aef683a9f8504edd
*******************************
a0(a11)=a11
a0(a11)=a0(a111)^a0(a110)^(!(a111^a110)va11)
a1=11111010010101111001111
!a0(a1)
*******************************
This is just my definition above encoded.
if THM is an axiom then THM is provable
if A is provable and B is provable
and A^B->THM
then THM is provable
proof( a11 ) <- proof( a111 ) & proof( a111 ) & not( a111 or a11 )
then
a11 = 8203215
not(proof(8203215)
Herc
Graham Cooper
May 20, 2013, 1:31:17 AM5/20/13
to
...
> a1 = 8203215
> not(proof(8203215)
>
The above formula refers to itself !
Herc
> a1 = 8203215
> not(proof(8203215)
>
The above formula refers to itself !
Herc
Charlie-Boo
May 20, 2013, 6:48:45 AM5/20/13
to
We have neither digits, numerals nor constant considerations. We have
only sets and functions. We have Q, defined to be (all A)(exists B)
LT(A,B) ^ P(B) ^ P(f(B)) where P be a recursive set and f(I) be a
recursive function. You have the recursive relation x proves y, the
r.e. set x is provable, the recursive function f(x)=y iff x proves y.
Let us pay homage to Godel, for He created the undecidable sentence
and decided it is true. How did he construct that temple from the
wisdom in his head? When will the Godel Sentence G be the same as the
C-B Sentence Q?
You must express "wff number x with x substituted for its free
variable is not provable", or, expanding your possibilities, realize
that,
"When any one of these sets [the true, provable and unrefutable
sentences] P, is expressible or representable, the sentence
that expresses or represents, respectively, 'This is in P.' is
undecidable." This includes:
1. Since unprovability is expressible: The sentence that expresses
"This is not provable." is undecidable.
2. Since refutability is expressible: The sentence that expresses
"This is refutable." is undecidable.
3. Since refutability is representable: The sentence that represents
"This is refutable." is undecidable.
http://www.cs.nyu.edu/pipermail/fom/2010-July/014933.html
When is each of these Q?
C-B
Graham Cooper
May 20, 2013, 5:21:22 PM5/20/13
to
On May 20, 8:48 pm, Charlie-Boo <shymath...@gmail.com> wrote:
> On May 20, 1:31 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > ...
>
> > > a1 = 8203215
> > > not(proof(8203215)
>
> > The above formula refers to itself !
>
> > Herc
>
> Yes, and a fine job indeed. But let us abstract UP and generalize.
> We have neither digits, numerals nor constant considerations. We have
> only sets and functions.
OK but you need SOME functionality in your parameters.
> On May 20, 1:31 am, Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> > ...
>
> > > a1 = 8203215
> > > not(proof(8203215)
>
> > The above formula refers to itself !
>
> > Herc
>
> Yes, and a fine job indeed. But let us abstract UP and generalize.
> We have neither digits, numerals nor constant considerations. We have
> only sets and functions.
Be it
call by name sqrt(X)
call by value sqrt(2)
call by line number, godel number..
20 not proof(20)
or
G<->not(proof(G))
are fine too, it depends what TYPE the domain of your functions are.
> We have Q, defined to be (all A)(exists B)
> LT(A,B) ^ P(B) ^ P(f(B)) where P be a recursive set and f(I) be a
> recursive function. You have the recursive relation x proves y, the
> r.e. set x is provable, the recursive function f(x)=y iff x proves y.
> Let us pay homage to Godel, for He created the undecidable sentence
> and decided it is true. How did he construct that temple from the
> wisdom in his head? When will the Godel Sentence G be the same as the
> C-B Sentence Q?
>
> You must express "wff number x with x substituted for its free
> variable is not provable", or, expanding your possibilities, realize
> that,
>
> "When any one of these sets [the true, provable and unrefutable
> sentences] P, is expressible or representable, the sentence
> that expresses or represents, respectively, 'This is in P.' is
> undecidable." This includes:
>
> 1. Since unprovability is expressible: The sentence that expresses
> "This is not provable." is undecidable.
> 2. Since refutability is expressible: The sentence that expresses
> "This is refutable." is undecidable.
> 3. Since refutability is representable: The sentence that represents
> "This is refutable." is undecidable.
>
> http://www.cs.nyu.edu/pipermail/fom/2010-July/014933.html
>
> When is each of these Q?
>
> C-B
GODEL = no PROVE(X) predicate
TARSKI = and no TRUE(X) either
isn't helping the poor old programmer!
As far as PROGRAMMING IN LOGIC is concerned.
2+1=3 ?
and
PROVE( 2+1=3 )
is just selecting the [TRACE=ON ] button!
----------------
That is why I simplified my provability predicate in Provable Set
Theory.
OLD
XeS<->p(X) <-> provable( XeS<->p(X) )
not(provable(X)) <-> provable(not(X))
provable(THM) <-> provable(A) ^ provable(B) ^ provable(A^B->THM)
-----------------
Godel was right in a way - YOU JUST CANT PROGRAM PROVABLE()!
it reduces down to the same as:
TRUE-WFF(X)
THEOREM(X)
DERIVE(X)
TRUE(X)
SOLVE(X)
THM(X)
T(X)
So THM(X) <-> NOT(PROVABLE(NOT(X)))
can just be done with double negation.
F<->F
~(F<->~F) %NO CONTRADICTIONS!
..
XeS<->p(X) <-> XeS<->p(X)
RUSSELLSET <-> (CONTRADICTION)
RUSSELLSET <-> FALSE
~EXIST(rs) rs<->x~ex
---------------------
I can't help any further because I simply don't have a predicate for
PROVE(THM)
t(R,z(Z)) :- if(L,R) , t(L,Z).
Where does it go?
Herc
Phil Carmody
May 23, 2013, 5:54:46 AM5/23/13
to
Pubkeybreaker <pubkey...@aol.com> writes:
> On May 17, 6:42ï¿œam, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> > In article <4c7758a5-2b88-4756-91f1-d59d43bcd...@d6g2000yqi.googlegroups.com>,
> >
> > Pubkeybreaker ￜ<pubkeybrea...@aol.com> wrote:
> > >This is a gross misstatement of the proof. It did NOT prove that there
> > >were infinitely many prime pairs. ï¿œWhat it did prove was that the gap
> > >between primes is FINITELY BOUNDED infinitely often. ï¿œThe bound is 70
as meaning that there is a finite gap such that there are infinitely many
consecutive primes with that gap. At no point did I interpret "prime pairs"
to mean "twin primes" rather than "consecutive primes a fixed, but unspecified,
distance apart".
Phil
--
"In a world of magnets and miracles"
-- Insane Clown Posse, Miracles, 2009. Much derided.
"Magnets, how do they work"
-- Pink Floyd, High Hopes, 1994. Lauded as lyrical geniuses.
> On May 17, 6:42ï¿œam, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> > In article <4c7758a5-2b88-4756-91f1-d59d43bcd...@d6g2000yqi.googlegroups.com>,
> >
> > Pubkeybreaker ￜ<pubkeybrea...@aol.com> wrote:
> > >This is a gross misstatement of the proof. It did NOT prove that there
> > >between primes is FINITELY BOUNDED infinitely often. ï¿œThe bound is 70
> > >x 10^6.
> >
> > I agree that the article (quoted from Scientific American) is unclear,
> > but it appears to be using "prime pairs" to mean "successive primes",
> > and "twin primes" to mean "prime pairs where the difference is 2".
> >
> > That makes the headline misleading because we already knew there were
> > infinitely many pairs of successive primes.
>
> Not with a finitely bounded gap between them we didn't.
I interpreted the headline (having already heard the story via other routes)
> >
> > I agree that the article (quoted from Scientific American) is unclear,
> > but it appears to be using "prime pairs" to mean "successive primes",
> > and "twin primes" to mean "prime pairs where the difference is 2".
> >
> > That makes the headline misleading because we already knew there were
> > infinitely many pairs of successive primes.
>
> Not with a finitely bounded gap between them we didn't.
as meaning that there is a finite gap such that there are infinitely many
consecutive primes with that gap. At no point did I interpret "prime pairs"
to mean "twin primes" rather than "consecutive primes a fixed, but unspecified,
distance apart".
Phil
--
"In a world of magnets and miracles"
-- Insane Clown Posse, Miracles, 2009. Much derided.
"Magnets, how do they work"
-- Pink Floyd, High Hopes, 1994. Lauded as lyrical geniuses.
Phil Carmody
May 23, 2013, 5:59:30 AM5/23/13
to
Aatu Koskensilta <aatu.kos...@uta.fi> writes:
> dull...@sprynet.com writes:
> > Some years ago there was something in Scientific American about
> > the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
My recollection is that it was Bill Gate in /The Road Ahead/ who
> dull...@sprynet.com writes:
> > Some years ago there was something in Scientific American about
> > the difficulty of factoring large primes.
>
> I once read it somewhere that factoring large primes is what is known
> as an NP ("Non-Polynomial") complete problem.
first mentioned how important an advance it would be if we could
factor large primes.
josh...@gmail.com
May 23, 2013, 6:41:38 AM5/23/13
to
I am not a mathematician - but can understand prime numbers, and even the hypothesis under discussion.
I wanted to tell to my children (who also know about prime numbers) about this development. Here is my script:
"For years mathematicians are struggling to prove that they will always find larger and larger cases of p where p and p+2 both are primes.
Someone recently proved that
if p is a prime number, within p + 70,000,000 there is another prime number q, no matter how large p is.
So, now mathematicians will work on finding what types of p's this 70 million is negotiable to smaller numbers, eventually going down to 2."
Is my understanding correct?
I wanted to tell to my children (who also know about prime numbers) about this development. Here is my script:
"For years mathematicians are struggling to prove that they will always find larger and larger cases of p where p and p+2 both are primes.
Someone recently proved that
if p is a prime number, within p + 70,000,000 there is another prime number q, no matter how large p is.
So, now mathematicians will work on finding what types of p's this 70 million is negotiable to smaller numbers, eventually going down to 2."
Is my understanding correct?
rossum
May 23, 2013, 8:01:49 AM5/23/13
to
On 23 May 2013 12:59:30 +0300, Phil Carmody
factor any large prime you give me. Unfortunately the margins of this
post are too small...
rossum
<thefatphi...@yahoo.co.uk> wrote:
>Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>> dull...@sprynet.com writes:
>> > Some years ago there was something in Scientific American about
>> > the difficulty of factoring large primes.
>>
>> I once read it somewhere that factoring large primes is what is known
>> as an NP ("Non-Polynomial") complete problem.
>
>My recollection is that it was Bill Gate in /The Road Ahead/ who
>first mentioned how important an advance it would be if we could
>factor large primes.
>
>Phil
Well, that's going to be a Fields Medal for me then. I can completely
>Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>> dull...@sprynet.com writes:
>> > Some years ago there was something in Scientific American about
>> > the difficulty of factoring large primes.
>>
>> I once read it somewhere that factoring large primes is what is known
>> as an NP ("Non-Polynomial") complete problem.
>
>My recollection is that it was Bill Gate in /The Road Ahead/ who
>first mentioned how important an advance it would be if we could
>factor large primes.
>
>Phil
factor any large prime you give me. Unfortunately the margins of this
post are too small...
rossum
quasi
May 23, 2013, 9:18:14 AM5/23/13
to
joshipura wrote:
>
>I am not a mathematician - but can understand prime numbers,
>and even the hypothesis under discussion.
>
>I wanted to tell to my children (who also know about prime
>numbers) about this development. Here is my script:
>
>"For years mathematicians are struggling to prove that they
>will always find larger and larger cases of p where p and p+2
>both are primes.
>
>Someone recently proved that if p is a prime number, within
>p + 70,000,000 there is another prime number q, no matter
>how large p is.
No.
>
>I am not a mathematician - but can understand prime numbers,
>and even the hypothesis under discussion.
>
>I wanted to tell to my children (who also know about prime
>numbers) about this development. Here is my script:
>
>"For years mathematicians are struggling to prove that they
>will always find larger and larger cases of p where p and p+2
>both are primes.
>
>Someone recently proved that if p is a prime number, within
>p + 70,000,000 there is another prime number q, no matter
>how large p is.
Let d = 70,000,000.
It's not true that for all primes p there is a prime q with
p < q <= p + d.
For example, let p be the largest prime less than (d + 1)! + 2
and let q be the least prime greater than p. Then q > p + d, so
there are no primes in the range p + 1 to p + d inclusive.
What was proved is that there are infinitely many primes pairs
p,q with p < q <= p + d.
quasi
Peter Percival
May 23, 2013, 8:47:03 AM5/23/13
to
josh...@gmail.com wrote:
> I am not a mathematician - but can understand prime numbers, and even the hypothesis under discussion.
>
> I wanted to tell to my children (who also know about prime numbers) about this development. Here is my script:
>
> "For years mathematicians are struggling to prove that they will always find larger and larger cases of p where p and p+2 both are primes.
>
> Someone recently proved that
> if p is a prime number, within p + 70,000,000 there is another prime number q, no matter how large p is.
There is a theorem that there is no upper limit on the number of
> I am not a mathematician - but can understand prime numbers, and even the hypothesis under discussion.
>
> I wanted to tell to my children (who also know about prime numbers) about this development. Here is my script:
>
> "For years mathematicians are struggling to prove that they will always find larger and larger cases of p where p and p+2 both are primes.
>
> Someone recently proved that
> if p is a prime number, within p + 70,000,000 there is another prime number q, no matter how large p is.
consecutive composites. Suppose you want at least n-1, (n >= 2)
consecutive composites, consider
n! + 2, n! + 3, n! + 4, ..., n! + n.
> So, now mathematicians will work on finding what types of p's this 70 million is negotiable to smaller numbers, eventually going down to 2."
>
> Is my understanding correct?
>
I think I am an Elephant,
Behind another Elephant
Behind /another/ Elephant who isn't really there....
A .A. Milne
David Bernier
May 23, 2013, 4:47:25 PM5/23/13
to
On 05/23/2013 05:59 AM, Phil Carmody wrote:
> Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>> dull...@sprynet.com writes:
>>> Some years ago there was something in Scientific American about
>>> the difficulty of factoring large primes.
>>
>> I once read it somewhere that factoring large primes is what is known
>> as an NP ("Non-Polynomial") complete problem.
>
> My recollection is that it was Bill Gate in /The Road Ahead/ who
> first mentioned how important an advance it would be if we could
> factor large primes.
>
> Phil
>
With respect to P =? NP, Scott Aaronson recently wrote:
> Aatu Koskensilta <aatu.kos...@uta.fi> writes:
>> dull...@sprynet.com writes:
>>> Some years ago there was something in Scientific American about
>>> the difficulty of factoring large primes.
>>
>> I once read it somewhere that factoring large primes is what is known
>> as an NP ("Non-Polynomial") complete problem.
>
> My recollection is that it was Bill Gate in /The Road Ahead/ who
> first mentioned how important an advance it would be if we could
> factor large primes.
>
> Phil
>
`` It�s just too hopelessly far beyond our current abilities[...]"
Ref.:
< http://www.scottaaronson.com/blog/?p=1385#comment-73266 > .
dave
--
99997066781489109195113098994290469881614963208468
1treePetrifiedForestLane
May 23, 2013, 5:58:22 PM5/23/13
to
that's like, how many femtoseconds per googolplex?
Brun's constant is not transcendental.
> > Is my understanding correct?
Brun's constant is not transcendental.
1treePetrifiedForestLane
May 23, 2013, 10:02:08 PM5/23/13
to
by hand, I recently showed that a)
there are even numbers not expressible as the sum
of twin primes; yay.
unless, I erred, theresville.
there are even numbers not expressible as the sum
of twin primes; yay.
unless, I erred, theresville.
josh...@gmail.com
May 24, 2013, 12:21:20 AM5/24/13
to
"For years mathematicians are struggling to prove that they will always find larger and larger cases of p where p and p+2 both are primes.
Someone recently proved that
**
Someone recently proved that
there are as many prime numbers p and q less than 70,000,000 apart as you want
**
Graham Cooper
May 24, 2013, 1:16:31 AM5/24/13
to
you're implying an algorithm exists such snd such...
which although true, it is not considered as being solved by
engineering standards
as O( |prime| ) = 10^|prime|
to actually find them incrementally (at will).
How many pairs < 70million apart have been found?
Herc
--
www.BLoCKPROLOG.com
quasi
May 24, 2013, 2:47:05 AM5/24/13
to
joshipura wrote:
>OK. So here goes changed script for review:
>
>"For years mathematicians are struggling to prove that they
>will always find larger and larger cases of p where p and p+2
>both are primes.
>
>Someone recently proved that
>**
>there are as many prime numbers p and q less than 70,000,000
>apart as you want
>**
Yes, In a sense.
>OK. So here goes changed script for review:
>
>"For years mathematicians are struggling to prove that they
>will always find larger and larger cases of p where p and p+2
>both are primes.
>
>Someone recently proved that
>**
>there are as many prime numbers p and q less than 70,000,000
>apart as you want
>**
The phrase "as many as you want" can be interpreted as
"infinitely many" (provided you always want more and more).
>So, now mathematicians will work on finding what types of
>p's this 70 million is negotiable to smaller numbers,
>eventually going down to 2."
the primes which have nearby successor primes.
They simply need to show, for a given integer d >= 2, that
there are infinitely prime pairs p,q with p < q <= p + d.
But yes, the goal is to keep reducing the value of d for which
they can prove the existence of infinitely many such prime pairs,
all the way down to d = 2, if possible.
quasi
Graham Cooper
May 24, 2013, 3:22:24 AM5/24/13
to
On May 24, 3:46 pm, quasi <qu...@null.set> wrote:
> joshipura wrote:
> >OK. So here goes changed script for review:
>
> >"For years mathematicians are struggling to prove that they
> >will always find larger and larger cases of p where p and p+2
> >both are primes.
>
> >Someone recently proved that
> >**
> >there are as many prime numbers p and q less than 70,000,000
> >apart as you want
> >**
>
> Yes, In a sense.
>
> The phrase "as many as you want" can be interpreted as
> "infinitely many" (provided you always want more and more).
>
Incorrect if you can't actually calculate as many as you want.
> joshipura wrote:
> >OK. So here goes changed script for review:
>
> >"For years mathematicians are struggling to prove that they
> >will always find larger and larger cases of p where p and p+2
> >both are primes.
>
> >Someone recently proved that
> >**
> >there are as many prime numbers p and q less than 70,000,000
> >apart as you want
> >**
>
> Yes, In a sense.
>
> The phrase "as many as you want" can be interpreted as
> "infinitely many" (provided you always want more and more).
>
Can you tell me a factor of any composite number?
3506641086599522334960321627880596993888147560566
70275244851438515265106048595338339402871505719094
41798207282164471551373680419703964191743046496589
27425623934102086438320211037295872576235850964311
05640735015081875106765946292055636855294752135008
52879416377328533906109750544334999811150056977236
890927563
Herc
Aatu Koskensilta
May 24, 2013, 11:28:25 AM5/24/13
to
In order to tell whether a given number is prime or composite it's not
necessary to factor the number.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
necessary to factor the number.
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Pubkeybreaker
May 24, 2013, 1:43:21 PM5/24/13
to
>
> - Show quoted text -
It's divisible by 13401
Robin Chapman
May 24, 2013, 2:36:24 PM5/24/13
to
On 24/05/2013 03:02, 1treePetrifiedForestLane wrote:
> by hand, I recently showed that a)
> there are even numbers not expressible as the sum
> of twin primes; yay.
>
> unless, I erred, theresville.
I'll bear that in mind.
> by hand, I recently showed that a)
> there are even numbers not expressible as the sum
> of twin primes; yay.
>
> unless, I erred, theresville.
Robin Chapman
May 24, 2013, 2:36:44 PM5/24/13
to
On 23/05/2013 22:58, 1treePetrifiedForestLane wrote:
> that's like, how many femtoseconds per googolplex?
>
> Brun's constant is not transcendental.
Here goes again ....
> that's like, how many femtoseconds per googolplex?
>
> Brun's constant is not transcendental.
1treePetrifiedForestLane
May 24, 2013, 5:36:10 PM5/24/13
to
I lost the piece of paper, or tossed it;
it was just over a hundred, I think.
> I'll bear that in mind.
as for Brun's, you don't have a thing against it,
especially the "heuristically derived" digits
that appeared in Ribbenboim's first edition
of "prime number records,: which is great intro.
obviously, a proof of necessity is desirable,
as it always to be desired for a fermatiste.
it was just over a hundred, I think.
> I'll bear that in mind.
especially the "heuristically derived" digits
that appeared in Ribbenboim's first edition
of "prime number records,: which is great intro.
obviously, a proof of necessity is desirable,
as it always to be desired for a fermatiste.
Graham Cooper
May 24, 2013, 5:38:58 PM5/24/13
to
On May 25, 1:28 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
My point is whether the proof is:
EXISTENCE of inf-many prime pairs
OR
CALCULATION METHOD of arbitrarily many prime pairs
----------------
Hence my question, how many prime pairs (over 70,000,000 say) have
been found?
if the answer is '3' then I don't think the 'find as many as you want'
clause is entirely accurate!
Herc
> Graham Cooper <grahamcoop...@gmail.com> writes:
> > Can you tell me a factor of any composite number?
>
> In order to tell whether a given number is prime or composite it's not
> necessary to factor the number.
>
>
It's easier but still difficult.
> > Can you tell me a factor of any composite number?
>
> In order to tell whether a given number is prime or composite it's not
> necessary to factor the number.
>
>
My point is whether the proof is:
EXISTENCE of inf-many prime pairs
OR
CALCULATION METHOD of arbitrarily many prime pairs
----------------
Hence my question, how many prime pairs (over 70,000,000 say) have
been found?
if the answer is '3' then I don't think the 'find as many as you want'
clause is entirely accurate!
Herc
David Bernier
May 24, 2013, 6:00:28 PM5/24/13
to
think has been shown:
For any integer n > 1, there are primes p and q with:
(i) p < q
(ii) 10^n < p
(iii) |p-q| <= 70,000,000.
"prime numbers come in pairs": cause of confusion.
dave
--
99997066781489109195113098994290469881614963208468
1treePetrifiedForestLane
May 24, 2013, 7:26:25 PM5/24/13
to
I was trying to see if it applied to prime quadruples (or
B_4) ... and I did prove it,
using a sloppy-clever method, knowing that
the post-fermatistes generally like it if
"it is true for sufficiently large (say) even numbers; so,
I began with 200, and immediately hit a prime
taht was not a sum of two quadruples ... or,
was it four ... just using the quadruples
that were listed on Wookypoopya.
B_4) ... and I did prove it,
using a sloppy-clever method, knowing that
the post-fermatistes generally like it if
"it is true for sufficiently large (say) even numbers; so,
I began with 200, and immediately hit a prime
taht was not a sum of two quadruples ... or,
was it four ... just using the quadruples
that were listed on Wookypoopya.
tucso...@me.com
May 24, 2013, 8:46:55 PM5/24/13
to
On Friday, May 24, 2013 2:38:58 PM UTC-7, Graham Cooper wrote:
>
> My point is whether the proof is:
>
> EXISTENCE of inf-many prime pairs
>
>
> OR
>
>
> CALCULATION METHOD of arbitrarily many prime pairs
>
>
I believe that you have misinterpreted the result.
>
> My point is whether the proof is:
>
> EXISTENCE of inf-many prime pairs
>
>
> OR
>
>
> CALCULATION METHOD of arbitrarily many prime pairs
>
>
The result says there are an infinite number of consecutive
primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
Two primes are "consecutive" when there are no primes in between
those primes.
Nobody has every shown there must exist a set of twin primes,
where p(n+1) - p(n) = 2, in any arbitrary range. That would prove
the Twin Prime Conjecture.
The task is now to reduce that range from 70,000,000 to 2.
With this clarified, I will give my response to your question.
Not that I know for sure, but....
I would think that the proof would be of the former.
It, probably, would simply say when we have have two primes
within 70,000,000 of each other, then we have two
greater primes within 70,000,000 of each other; or else
we would have some contradiction.
That is, I believe the proof would be non-constructive.
Again, I don't "know". This is speculation.
However, the proof may be constructive enough that
it gives a range as to where those greater primes occur,
then we could, in theory, check that range to find them.
Of course, this range will have to be very large, and will
take a very long time to check all numbers in there.
>
>
> ----------------
>
>
>
> Hence my question, how many prime pairs (over 70,000,000 say) have
>
> been found?
>
>
>
> if the answer is '3' then I don't think the 'find as many as you want'
>
> clause is entirely accurate!
>
70,000,000, there are plenty of examples.
2 and 3,
3 and 5,
5 and 7,
...,
29 and 31,
31 and 37,
...
8867 and 8887,
and every two consecutive primes less than 70,000,000.
And more.
And according to the proof, an infinite number.
As for very very very large instances of such primes:
I haven't found that they have been calculate those
prime, hence I suspect the proof doesn't construct them
or they are so large that it is difficult to churn the algorithms.
>
> Herc
ZG
Message has been deleted
Graham Cooper
May 24, 2013, 9:40:09 PM5/24/13
to
On May 25, 10:46 am, "Tucsond...@me.com" <tucsond...@me.com> wrote:
> On Friday, May 24, 2013 2:38:58 PM UTC-7, Graham Cooper wrote:
>
> > My point is whether the proof is:
>
> > EXISTENCE of inf-many prime pairs
>
> > OR
>
> > CALCULATION METHOD of arbitrarily many prime pairs
>
> I believe that you have misinterpreted the result.
>
> The result says there are an infinite number of consecutive
> primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
> Two primes are "consecutive" when there are no primes in between
> those primes.
No it doesn't. We just proved an arbitrary size succession
> On Friday, May 24, 2013 2:38:58 PM UTC-7, Graham Cooper wrote:
>
> > My point is whether the proof is:
>
> > EXISTENCE of inf-many prime pairs
>
> > OR
>
> > CALCULATION METHOD of arbitrarily many prime pairs
>
> I believe that you have misinterpreted the result.
>
> The result says there are an infinite number of consecutive
> primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
> Two primes are "consecutive" when there are no primes in between
> those primes.
of composites 10 posts ago!
n!-n .. n!-2
OK - GIVE ME 100 of them over 70,000,000 !
For any integer n > 1, there are primes p and q with:
(i) p < q
(ii) 10^n < p
(iii) |p-q| <= 70,000,000.
2 primes > 100
2 primes > 1000
2 primes > 10000
2 primes > 100000
That is NOT AS MANY AS I WANT!
FFS! what morons.
Herc
Herc
David Bernier
May 24, 2013, 9:48:03 PM5/24/13
to
"Bounded gaps between primes" (I din't choose it...)
with a summary here:
http://annals.math.princeton.edu/articles/7954
dave
--
99997066781489109195113098994290469881614963208468
Graham Cooper
May 24, 2013, 9:51:56 PM5/24/13
to
prime1 > 100000000000000000000000000000000000000000
prime2 < 700000000000000000000000000000000000000000000000
Herc
Herc
Graham Cooper
May 24, 2013, 9:56:40 PM5/24/13
to
primepair-50b < 70000000 + primepair-50a
Herc
Graham Cooper
May 24, 2013, 10:08:25 PM5/24/13
to
MATHEMATICIAN: as many as you want!
COMPUTER SCIENTIST: not by Engineering standards!
MATHEMATICIAN: infinitely many!
COMPUTER SCIENTIST: the time complexity is exponential
MATHEMATICIAN: as many as you want = infinitely many
COMPUTER SCIENTIST: the sizes go up too rapidly 10^n
MATHEMATICIAN: you get as many as you want!
COMPUTER SCIENTIST:
1st is size 10
2nd is size 100
3rd is size 1000
4th is size 10000
5th is size 100000
MATHAMATICAN: that's as many as you want, check the article!
COMPUTER SCIENTIST: the 50TH result is bigger than
10000000000000000000000000000000000000000000000000
MATHEMATICIAN: oh...
------------
Herc
--
www.BLoCKPROLOG.com
COMPUTER SCIENTIST: not by Engineering standards!
MATHEMATICIAN: infinitely many!
COMPUTER SCIENTIST: the time complexity is exponential
MATHEMATICIAN: as many as you want = infinitely many
COMPUTER SCIENTIST: the sizes go up too rapidly 10^n
MATHEMATICIAN: you get as many as you want!
COMPUTER SCIENTIST:
1st is size 10
2nd is size 100
3rd is size 1000
4th is size 10000
5th is size 100000
MATHAMATICAN: that's as many as you want, check the article!
COMPUTER SCIENTIST: the 50TH result is bigger than
10000000000000000000000000000000000000000000000000
MATHEMATICIAN: oh...
------------
Herc
--
www.BLoCKPROLOG.com
tucso...@me.com
May 24, 2013, 11:07:25 PM5/24/13
to
On Friday, May 24, 2013 6:40:09 PM UTC-7, Graham Cooper wrote:
> On May 25, 10:46 am, "Tucsond...@me.com" <tucsond...@me.com> wrote:
>
> > On Friday, May 24, 2013 2:38:58 PM UTC-7, Graham Cooper wrote:
>
> >
>
> > > My point is whether the proof is:
>
> >
>
> > > EXISTENCE of inf-many prime pairs
>
> >
>
> > > OR
>
> >
>
> > > CALCULATION METHOD of arbitrarily many prime pairs
>
> >
>
> > I believe that you have misinterpreted the result.
>
> >
>
> > The result says there are an infinite number of consecutive
>
> > primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
>
> > Two primes are "consecutive" when there are no primes in between
>
> > those primes.
>
>
>
>
>
>
>
> No it doesn't. We just proved an arbitrary size succession
>
> of composites 10 posts ago!
>
>
>
> n!-n .. n!-2
>
>
But you never proved that any gap occurs I finely many times.
> On May 25, 10:46 am, "Tucsond...@me.com" <tucsond...@me.com> wrote:
>
> > On Friday, May 24, 2013 2:38:58 PM UTC-7, Graham Cooper wrote:
>
> >
>
> > > My point is whether the proof is:
>
> >
>
> > > EXISTENCE of inf-many prime pairs
>
> >
>
> > > OR
>
> >
>
> > > CALCULATION METHOD of arbitrarily many prime pairs
>
> >
>
> > I believe that you have misinterpreted the result.
>
> >
>
> > The result says there are an infinite number of consecutive
>
> > primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
>
> > Two primes are "consecutive" when there are no primes in between
>
> > those primes.
>
>
>
>
>
>
>
> No it doesn't. We just proved an arbitrary size succession
>
> of composites 10 posts ago!
>
>
>
> n!-n .. n!-2
>
>
>
> For any integer n > 1, there are primes p and q with:
>
> (i) p < q
>
> (ii) 10^n < p
>
> (iii) |p-q| <= 70,000,000.
>
>
>
>
>
> that 2 primes >10
>
> 2 primes > 100
>
> 2 primes > 1000
>
> 2 primes > 10000
>
> 2 primes > 100000
>
>
>
>
>
> That is NOT AS MANY AS I WANT!
>
>
>
>
>
> FFS! what morons.
>
>
> Herc
>
>
>
>
>
>
>
>
>
> Herc
Twotever, ZG
Message has been deleted
Graham Cooper
May 24, 2013, 11:11:27 PM5/24/13
to
This is NOTHING TO DO
WITH ANY 'SUPPORTING' POSTS SINCE MADE ABOUT THAT STATEMENT.
---------
POST 1000!
your final 2 numbers should be 1000 digits long!
Herc
Graham Cooper
May 24, 2013, 11:13:15 PM5/24/13
to
> > > The result says there are an infinite number of consecutive
>
> > > primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
>
> > > Two primes are "consecutive" when there are no primes in between
>
> > > those primes.
>
> > No it doesn't. We just proved an arbitrary size succession
>
> > of composites 10 posts ago!
>
> > n!-n .. n!-2
>
> But you never proved that any gap occurs I finely many times.
>
>
FOR ALL N
>
> > > primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
>
> > > Two primes are "consecutive" when there are no primes in between
>
> > > those primes.
>
> > No it doesn't. We just proved an arbitrary size succession
>
> > of composites 10 posts ago!
>
> > n!-n .. n!-2
>
> But you never proved that any gap occurs I finely many times.
>
>
There is nothing consecutive going on, not in this newsgroup.
Herc
Zeit Geist
May 25, 2013, 1:47:46 AM5/25/13
to
> Phil
>
> --
>
> "In a world of magnets and miracles"
>
> -- Insane Clown Posse, Miracles, 2009. Much derided.
>
> "Magnets, how do they work"
>
> -- Pink Floyd, High Hopes, 1994. Lauded as lyrical geniuses.
The second quote is actually from High Hopes.
I'm not sure about the second one.
ZG
Richard Tobin
May 25, 2013, 4:57:05 AM5/25/13
to
In article <c948ab01-3de4-4753...@kt20g2000pbb.googlegroups.com>,
-- Richard
Graham Cooper <graham...@gmail.com> wrote:
>> The result says there are an infinite number of consecutive
>> primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
>> Two primes are "consecutive" when there are no primes in between
>> those primes.
>No it doesn't. We just proved an arbitrary size succession
>of composites 10 posts ago!
You have misunderstood. The two statements are not contradictory.
>> The result says there are an infinite number of consecutive
>> primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
>> Two primes are "consecutive" when there are no primes in between
>> those primes.
>No it doesn't. We just proved an arbitrary size succession
>of composites 10 posts ago!
-- Richard
sperm...@yahoo.com
May 25, 2013, 5:07:13 AM5/25/13
to
On Friday, May 17, 2013 2:03:24 AM UTC+10, Sam Wormley wrote:
> First Proof That Infinitely Many Prime Numbers Come in Pairs
>
> > http://www.scientificamerican.com/article.cfm?id=first-proof-that-infinite-many-prime-numbers-come-in-pairs
>
>
>
> > That goal is the proof to a conjecture concerning prime numbers.
>
> > Those are the whole numbers that are divisible only by one and
>
> > themselves. Primes abound among smaller numbers, but they become less
>
> > and less frequent as one goes towards larger numbers. In fact, the
>
> > gap between each prime and the next becomes larger and larger -- on
>
> > average. But exceptions exist: the 'twin primes', which are pairs of
>
> > prime numbers that differ in value by 2. Examples of known twin
>
> > primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2^195,000 - 1 and
>
> > 2,003,663,613 × 2^195,000 + 1.
>
> >
>
> > The twin prime conjecture says that there is an infinite number of
>
> > such twin pairs. Some attribute the conjecture to the Greek
>
> > mathematician Euclid of Alexandria, which would make it one of the
>
> > oldest open problems in mathematics.
All talk about prime number is meaningless as mathematician dont even know what a number is- without circularity
all their definitions about numbers reduce to just this
a number is a number-circularity impredicative
thus we then dont know what a number is
mathematicians give all these proofs about prime numbers but they dont even know what a number is so their proofs are worthless
as without knowing what a number is they then cant even IDENTIFY what a prime number is
Australias lead erotic poet colin leslie dean points out Mathematicians cannot define a number with out being impredicative-ie self referential thus mathematicians dont even know what a number is- thus maths is meaningless All mathematicians can say is a number is a number ?thus they don?t know what a number is thus maths is meaningless
http://www.scribd.com/doc/40697621/Mathematics-Ends-in-Meaninglessness-ie-self-contradiction
http://www.iep.utm.edu/predicat/
http://www.iep.utm.edu/predicat/
In many approaches to the foundations of mathematics, the property N
of being a natural number is defined as follows. An object x has the
property N just in case x has every property F which is had by zero
and is inherited from any number u to its successor u+1. Or in
symbols:
Def-N N(x) ? ?F[F(0) ? ?u(F(u) ? F(u + 1)) ? F(x)]
This definition has the nice feature of entailing the principle of
mathematical induction, which says that any property F which is had by
zero and is inherited from any number u to its successor u+1 is had by
every natural number:
?F{F(0) ? ?u(F(u) ? F(u + 1)) ? ?x(N(x) ? F(x))}
However, Def-N is impredicative because it defines the property N by
generalizing over all arithmetical properties, including the one being
defined.
again impredicative definition
Let n be smallest natural number such that every natural number can be
written as the sum of at most four cubes.
again impredicative definition
http://en.wikipedia.org/wiki/Impredicativity
Concerning mathematics, an example of an impredicative definition is
the smallest number in a set, which is formally defined as: y = min(X)
if and only if for all elements x of X, y is less than or equal to x,
and y is in X.
http://en.wikipedia.org/wiki/Set-theore ... al_numbers
http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
A consequence of Kurt Gödel's work on incompleteness is that in any effectively generated axiomatization of number theory (ie. one containing minimal arithmetic), there will be true statements of number theory which cannot be proven in that system. So trivially it follows that ZFC or any other effectively generated formal system cannot capture entirely what a number is.
Whether this is a problem or not depends on whether you were seeking a formal definition of the concept of number. For people such as Bertrand Russell (who thought number theory, and hence mathematics, was a branch of logic and number was something to be defined in terms of formal logic) it was an insurmountable problem. But if you take the concept of number as an absolutely fundamental and irreducible one, it is to be expected. After all, if any concept is to be left formally undefined in mathematics, it might as well be one which everyone understands.
Poincaré, amongst others (Bernays, Wittgenstein), held that any attempt to define natural number as it is endeavoured to do so above is doomed to failure by circularity. Informally, Gödel's theorem shows that a formal axiomatic definition is impossible (incompleteness), Poincaré claims that no definition, formal or informal, is possible (circularity). As such, they give two separate reasons why purported definitions of number must fail to define number. A quote from Poincaré: "The definitions of number are very numerous and of great variety, and I will not attempt to enumerate their names and their authors. We must not be surprised that there are so many. If any of them were satisfactory we should not get any new ones." A quote from Wittgenstein: "This is not a definition. This is nothing but the arithmetical calculus with frills tacked on." A quote from Bernays: "Thus in spite of the possibility of incorporating arithmetic into logistic, arithmetic constitutes the more abstract ('purer') schema; and this appears paradoxical only because of a traditional, but on closer examination unjustified view according to which logical generality is in every respect the highest generality."
> First Proof That Infinitely Many Prime Numbers Come in Pairs
>
> > http://www.scientificamerican.com/article.cfm?id=first-proof-that-infinite-many-prime-numbers-come-in-pairs
>
>
>
> > That goal is the proof to a conjecture concerning prime numbers.
>
> > Those are the whole numbers that are divisible only by one and
>
> > themselves. Primes abound among smaller numbers, but they become less
>
> > and less frequent as one goes towards larger numbers. In fact, the
>
> > gap between each prime and the next becomes larger and larger -- on
>
> > average. But exceptions exist: the 'twin primes', which are pairs of
>
> > prime numbers that differ in value by 2. Examples of known twin
>
> > primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2^195,000 - 1 and
>
> > 2,003,663,613 × 2^195,000 + 1.
>
> >
>
> > The twin prime conjecture says that there is an infinite number of
>
> > such twin pairs. Some attribute the conjecture to the Greek
>
> > mathematician Euclid of Alexandria, which would make it one of the
>
> > oldest open problems in mathematics.
All talk about prime number is meaningless as mathematician dont even know what a number is- without circularity
all their definitions about numbers reduce to just this
a number is a number-circularity impredicative
thus we then dont know what a number is
mathematicians give all these proofs about prime numbers but they dont even know what a number is so their proofs are worthless
as without knowing what a number is they then cant even IDENTIFY what a prime number is
Australias lead erotic poet colin leslie dean points out Mathematicians cannot define a number with out being impredicative-ie self referential thus mathematicians dont even know what a number is- thus maths is meaningless All mathematicians can say is a number is a number ?thus they don?t know what a number is thus maths is meaningless
http://www.scribd.com/doc/40697621/Mathematics-Ends-in-Meaninglessness-ie-self-contradiction
http://www.iep.utm.edu/predicat/
http://www.iep.utm.edu/predicat/
In many approaches to the foundations of mathematics, the property N
of being a natural number is defined as follows. An object x has the
property N just in case x has every property F which is had by zero
and is inherited from any number u to its successor u+1. Or in
symbols:
Def-N N(x) ? ?F[F(0) ? ?u(F(u) ? F(u + 1)) ? F(x)]
This definition has the nice feature of entailing the principle of
mathematical induction, which says that any property F which is had by
zero and is inherited from any number u to its successor u+1 is had by
every natural number:
?F{F(0) ? ?u(F(u) ? F(u + 1)) ? ?x(N(x) ? F(x))}
However, Def-N is impredicative because it defines the property N by
generalizing over all arithmetical properties, including the one being
defined.
again impredicative definition
Let n be smallest natural number such that every natural number can be
written as the sum of at most four cubes.
again impredicative definition
http://en.wikipedia.org/wiki/Impredicativity
Concerning mathematics, an example of an impredicative definition is
the smallest number in a set, which is formally defined as: y = min(X)
if and only if for all elements x of X, y is less than or equal to x,
and y is in X.
http://en.wikipedia.org/wiki/Set-theore ... al_numbers
http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers
A consequence of Kurt Gödel's work on incompleteness is that in any effectively generated axiomatization of number theory (ie. one containing minimal arithmetic), there will be true statements of number theory which cannot be proven in that system. So trivially it follows that ZFC or any other effectively generated formal system cannot capture entirely what a number is.
Whether this is a problem or not depends on whether you were seeking a formal definition of the concept of number. For people such as Bertrand Russell (who thought number theory, and hence mathematics, was a branch of logic and number was something to be defined in terms of formal logic) it was an insurmountable problem. But if you take the concept of number as an absolutely fundamental and irreducible one, it is to be expected. After all, if any concept is to be left formally undefined in mathematics, it might as well be one which everyone understands.
Poincaré, amongst others (Bernays, Wittgenstein), held that any attempt to define natural number as it is endeavoured to do so above is doomed to failure by circularity. Informally, Gödel's theorem shows that a formal axiomatic definition is impossible (incompleteness), Poincaré claims that no definition, formal or informal, is possible (circularity). As such, they give two separate reasons why purported definitions of number must fail to define number. A quote from Poincaré: "The definitions of number are very numerous and of great variety, and I will not attempt to enumerate their names and their authors. We must not be surprised that there are so many. If any of them were satisfactory we should not get any new ones." A quote from Wittgenstein: "This is not a definition. This is nothing but the arithmetical calculus with frills tacked on." A quote from Bernays: "Thus in spite of the possibility of incorporating arithmetic into logistic, arithmetic constitutes the more abstract ('purer') schema; and this appears paradoxical only because of a traditional, but on closer examination unjustified view according to which logical generality is in every respect the highest generality."
David Bernier
May 25, 2013, 5:09:05 AM5/25/13
to
Then the numbers
d-101, d-100, d-99, d-98, ... d-2 are all composites.
That gives (at least) 100 consecutive composites.
Using my trusty computer and PARI/gp, I find that
both d + 101,081 and d + 101,083 are prime numbers.
So there's a gap of size exactly 2 in the primes beyond
101! but before 101! + 200,000.
(101!+101081, 101!+101083) is a twin primes "pair".
One of the long term goals of researchers on
"small" gaps between consecutive primes would
be to prove the Twin Prime Conjecture.
The new result says that gaps of size
less than 70,000,000 happen infinitely often.
It's a very theoretical proof.
Are you a hyperfinitist? There's a proof that
"There are infinitely many primes." and also of
"Every composite natural n > 1 can be expressed
as a product of primes." .
Those results are also theoretical, and don't give efficient
tricks and algorithms to find large primes explicitly,
or to help in factoring a 300-digit composite
number into a product of primes.
dave
? isprime( prod(X=1,101, X) + 101081)
%408 = 1
? isprime( prod(X=1,101, X) + 101083)
%409 = 1
--
99997066781489109195113098994290469881614963208468
1treePetrifiedForestLane
May 25, 2013, 3:34:17 PM5/25/13
to
bad math eroticism
Graham Cooper
May 25, 2013, 10:09:24 PM5/25/13
to
On May 25, 6:57 pm, rich...@cogsci.ed.ac.uk (Richard Tobin) wrote:
> In article <c948ab01-3de4-4753-8182-c53b2531e...@kt20g2000pbb.googlegroups.com>,
n is SOME IDENTIFIER INDEX out of nowhere...
Not true for all n.
Would read better as p_a and p_b
or p_a(n) and p_b(n)
by merely moving 'such that' a little
The result says there are an infinite number of consecutive
primes such that p(n) and p(n+1) have p(n+1) - p(n) < 70,000,000.
Worse than 'as many as you want'
If someone asks for a ACCURATE NEWSPAPER SUBJECT LINE
then don't feed them bullshit
then repeatedly misconstrue the objections to why it is incorrect.
You MATHNOS have been LYING THROUGH YOUR TEETH for 10 YEARS
| R | > | INFINITE LIST ROWS |
proof: 0.4444444454444444544444444445544444444444544445444444..
You're all full of sh!t, and you'll stay that way I'm not helping any
more when your vomitful antics from the same gang of cowards never
stops.
Herc
> In article <c948ab01-3de4-4753-8182-c53b2531e...@kt20g2000pbb.googlegroups.com>,
> Graham Cooper <grahamcoop...@gmail.com> wrote:
>
> >> The result says there are an infinite number of consecutive
> >> primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
> >> Two primes are "consecutive" when there are no primes in between
> >> those primes.
> >No it doesn't. We just proved an arbitrary size succession
> >of composites 10 posts ago!
>
> You have misunderstood. The two statements are not contradictory.
>
> -- Richard
OK, but that has more ambiguous wording than Anastasia.
>
> >> The result says there are an infinite number of consecutive
> >> primes p(n) and p(n+1), such that p(n+1) - p(n) < 70,000,000.
> >> Two primes are "consecutive" when there are no primes in between
> >> those primes.
> >No it doesn't. We just proved an arbitrary size succession
> >of composites 10 posts ago!
>
> You have misunderstood. The two statements are not contradictory.
>
> -- Richard
n is SOME IDENTIFIER INDEX out of nowhere...
Not true for all n.
Would read better as p_a and p_b
or p_a(n) and p_b(n)
by merely moving 'such that' a little
The result says there are an infinite number of consecutive
Worse than 'as many as you want'
If someone asks for a ACCURATE NEWSPAPER SUBJECT LINE
then don't feed them bullshit
then repeatedly misconstrue the objections to why it is incorrect.
You MATHNOS have been LYING THROUGH YOUR TEETH for 10 YEARS
| R | > | INFINITE LIST ROWS |
proof: 0.4444444454444444544444444445544444444444544445444444..
You're all full of sh!t, and you'll stay that way I'm not helping any
more when your vomitful antics from the same gang of cowards never
stops.
Herc
Robin Chapman
May 28, 2013, 6:01:00 AM5/28/13
to
On 24/05/2013 22:36, 1treePetrifiedForestLane wrote:
> I lost the piece of paper, or tossed it;
So maybe you're a tosser ?
> I lost the piece of paper, or tossed it;
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