Twin Primes Conjecture

[wpolly]

A further generalization:
for every k-tuple of irreducible polynomials P1, P2, ... Pn with no congrurence contradictions(for example not n and n^2+2 because one of them will always be divisible by 3) there'll be infinitely many k such that P1(k), P2(k), ...., Pn(k) are all primes.

[philmoore]

There is another message at:
http://listserv.nodak.edu/scripts/wa...406&L=nmbrthry
claiming that lemma 8 is wrong as stated.

[jinydu]

http://mathworld.wolfram.com/news/20...09/twinprimes/

"Twin Prime Proof Proffered
By Eric W. Weisstein
June 9, 2004--A recent preprint by Vanderbilt University mathematician R. F. Arenstorf appears to come close to settling the longstanding question of the infinitude of twin primes. Twin primes are pairs of prime numbers such that the larger member of the pair is exactly 2 greater than the smaller, i.e., primes p and q such that q - p = 2. Explicitly, the first few twin primes are (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ....

The properties and distribution of twin primes (first given this name by Paul Stäckel, 1892-1919), are active areas of mathematical research. While the distribution of twin primes has remained elusive, mathematician V. Brun proved in 1919 that the sum of the reciprocals of the members of each twin prime pair



converges to a definite number even if the sum contains an infinite number a terms, a result known as Brun's theorem. The number B, known as Brun's constant, is difficult to compute, but is known to be approximately equal to 1.902160583104. (Amusingly, it was Thomas Nicely's 1995 high-precision computation of Brun's constant that first revealed a serious hardware bug in Intel's Pentium microprocessor.) Since the sum of the reciprocals of all the primes diverges (which represents a strengthening of Euclid's second theorem on the infinitude of the primes that was first proved by Euler in 1737), Brun's theorem shows that the twin primes are sparsely distributed among the primes.

The twin prime conjecture states that there are an infinite number of twin primes. While Hardy and Wright (1979, p. 5) note that "the evidence, when examined in detail, appears to justify the conjecture," and Shanks (1993, p. 219) states even more strongly, "the evidence is overwhelming," Hardy and Wright also note that the proof or disproof of conjectures of this type "is at present beyond the resources of mathematics."

In fact, no proof of the twin primes conjecture has been constructed despite the efforts of dozens of mathematicians over almost a century. In contrast, a recent preprint has apparently succeeded in showing the existence of prime arithmetic progressions of any length k, a related and also long-outstanding problem (MathWorld headline news story, April 12, 2004).

In a May 26 preprint, R. F. Arenstorf published a proposed proof of the twin prime conjecture, even in a stronger form due to Hardy and Littlewood (1923). The proof uses methods from classical analytic number theory, including the properties of the Riemann zeta function, ideas from the proof of the prime number theorem, and a so-called Tauberian theorem due to Wiener and Ikehara dating back to 1931, the latter of which leads almost immediately to Arenstorf's main result.

While Arenstorf's approach looks promising, an error in one particular step of the proof (specifically, Lemma 8 on page 35, where a lemma is short theorem used in proving a larger theorem) has recently been pointed out by French mathematician Gérald Tenenbaum of the Institut Élie Cartan in Nancy (Tenenbaum 2004). While mathematicians remain hopeful that any holes in the proof can be corrected, Tenenbaum opines that this particular error may have serious consequences for the integrity of the overall proof. Additional analysis by other mathematicians over the coming weeks and months will establish if, like the originally flawed proof of Fermat's last theorem, the twin prime result can also be corrected, thus finally settling this long-open problem, or if it requires additional insight and tools before it can finally be cracked."

[mfgoode]

Originally Posted by ET_

If there are infinitely many, it's only a mater of time to find them out


Luigi

Here is a paraphrase from the book 'Archimedes' Revenge' by Paul Hoffman.
'What about primes-called twin primes- that differ by two?
Among the first 25 primes are 8 pairs of twin primes ----
For almost 150 yrs, number theorists have conjectured that pairs of twin primes are inexhaustable, like the primes themselves, but no one has been able to prove this.

Progres was made in 1966 when the Chinese math'cian Chen Jing-run proved that there exist infinitely many pairs of numbers that differ by two in whch the fiirst number is a prime and the second is either a prime or the product of two primes( these are called 'almost prime')

Chen proved a weaker version of Goldbachs conjecture: every "sufficiently large" even number is the sum of a prime and an 'almost prime' ".
Mathem'cians consider Chen's proof to be the most significant contribution to prime number theory in the last few decades'
Mally