Twin prime conjecture
There are an infinite number of primes p such that p + 2 is also prime.
Such a pair of prime numbers is called a twin prime. The conjecture has been researched by many number theorists. Mathematicians believe the conjecture to be true, based only on numerical evidence and heuristic reasoning involving the probabilistic distribution of primes.
In 1940, Erdös showed that there is a constant c < 1 and infinitely many primes p such that p' − p < c ln p, where p' denotes the next prime after p. This result was successively improved; in 1986 Maier showed that a constant c < 0.25 can be used.
In 1966, Chen Jingrun showed that there are infinitely many primes p such that p + 2 is either a prime or a semiprime (i.e., the product of two primes). The approach he took involved a topic called sieve theory, and he managed to treat the twin prime conjecture and Goldbach's conjecture in similar manners.
There is also a generalization of the twin prime conjecture, known as the Hardy-Littlewood conjecture (after G. H. Hardy and John Littlewood), which is concerned with the distribution of twin primes, in analogy to the prime number theorem. Let π2(x) denote the number of primes p ≤ x such that p + 2 is also prime. Define the twin prime constant C2 as
(here the product extends over all prime numbers p ≥ 3). Then the conjecture is that
in the sense that the quotient of the two expressions tends to 1 as x approaches infinity.
This conjecture can be justified (but not proven) by assuming that 1 / ln t describes the density function of the prime distribution, an assumption suggested by the prime number theorem. The numerical evidence behind the Hardy-Littlewood conjecture is quite impressive.
Serious problem found in potential proof
On May 26, 2004, Richard Arenstorf of Vanderbilt University submitted a 38-page proof that there are, in fact, infinitely many twin primes. On June 3, Michel Balazard of University Bordeaux reported that Lemma 8 on page 35 is false. As is typical in mathematical proofs, the defect may be correctable or a substitute method may repair or replace the defect. Arenstorf withdrew his proof on June 8, noting "A serious error has been found in the paper, specifically, Lemma 8 is incorrect".
There Are Infinitely Many Prime Twins - R. Arenstorf Withdrawn June 8, 2004.