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Twin Prime Conjecture Proven?

By kroth
Posted Sat May 29, 2004 at 01:10:42 PM PDT

There is an article on slashdot discussing a preprint of a paper by R. F. Arenstorf proving the twin primes conjecture.
Read about it here.
The discussion that follows also has some amusing points, but not a whole lot of math.

Of course you could also skip the article and just look at the preprint on arxiv here.

Update [2004-6-10 13:39:44 by overconvergent]: The preprint has been withdrawn from the arXiv; because:

A serious error has been found in the paper, specifically, Lemma 8 is incorrect.

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Twin Prime Conjecture Proven? | 5 comments (4 topical, 0 hidden)

[new] There seems to be some doubt emerging ... (none / 0) (#1)
by overconvergent on Sun Jun 06, 2004 at 05:37:41 PM PDT

on sci.math, the following comment was posted, by Zbigniew Fiedorowicz:

Michael Balazard of the University of Bordeaux has found a serious error in the proof:

> J'ai malheureusement trouv une erreur grave dans l'article d'Arenstorf.
> Le lemme 8, page 35 est manifestement faux, et il est fondamental. Il
> est possible que la dmonstration puisse tre rpare, mais c'est non
> trivial.

[new] TWIN PRIMES CONJECTURE (none / 0) (#2)
by Anonymous Hero on Mon May 16, 2005 at 08:54:54 AM PDT

Why do we need 38 pages to prove something which we can see quite intuitively and demonstrate easily in a page? I challenge anyone to refute the following (bearing in mind that the assumption of randomness is made):- The Twin Primes Conjecture is quite obviously seen to be true if we utilise 2 proven facts and one very reasonable assumption: * The primes are infinite in number * Bertrand's postulate (there is a prime in every interval [x, 2x] * We are prepared to accept the "randomness of the occurrence of primes within intervals". By this I mean that in any randomly selected interval [x, 2x], each integer in the interval has an equal likelihood (probability) of being prime. So lets proceed with our intuitive "proof". Consider the infinite array of intervals formed by the primes and their doubles: [p(m), 2p(m)], [p(m+1), 2p(m+1)], [p(m+2), 2p(m+2)], ... where p(m+k) is the (m+k)th prime such that P(m) < p(m+k), and k = 0,1,2,3, ... By Bertrand's postulate: the probability of any integer (2 + p(m+k) ) being prime is precisely 1/p(m+k) since at least one prime must exist in any interval [ p(m+k), 2p(m+k) ], and every integer in the interval has the equal probability, 1/p(m+k), of being prime. Now let us consider the largest known twin prime to be p(n), say. Let p(m) be some singleton prime such that p(n) < p(m). The probability that no further integer, p(m+k) + 2, is prime is precisely the infinite product (1-1/p(m+1)).(1-1/p(m+2)).(1 - 1/p(m+3))... = 0 Hence the probability that at least one of the intervals [p(m),2p(m)],[p(m+1),2p(m+1)], [p(m+2),2p(m+2)], ... contains a prime integer of form 2 + p(m+k), is exactly 1. Now, call this new twin prime, p(n') and repeat the above argument. Plainly, there will always be shown to be some interval greater than the largest known twin prime (p(n), p(n'), ... etc.) which contains a twin prime of similar form to 2 + p(m+k). Our intuitive proof based on the assumption of randomness is complete.

There are intuitive arguments for Goldbach ... by overconvergent, 05/18/2005 15:19:36 PDT (none / 0)

[new] Proofs of the Prime Twin Conjecture Using 'Random' (none / 0) (#5)
by mwiner on Tue Mar 20, 2007 at 09:35:17 PM PDT

In this thread a user raises a discussion of intuitive proofs of the prime twin conjecture. I have a proof which spells out why the primes are random, and why there are infinitely many prime twins (and other allowable constellations). http://www.rankyouragent.com/primes/primes.htm The user "anonymous hero" who posted the discussion regarding this style of proof is welcome to contact me.

Twin Prime Conjecture Proven? | 5 comments (4 topical, 0 hidden)

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