Twin prime conjecture
- There are infinitely many 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 1849 de Polignac made the more general conjecture that for every natural number k, there are infinitely many prime pairs p and p′ such that p - p′ = 2k. The case k = 1 is the twin prime conjecture.
In 1915, Viggo Brun showed that the sum of reciprocals of the twin primes was convergent. This famous result was the first use of the Brun sieve and helped initiate the development of modern sieve theory. The modern version of Brun's argument can be used to show that the number of twin primes less than N does not exceed
for some absolute constant C > 0.
In 1940, Paul Erdős 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 Helmut Maier showed that a constant c < 0.25 can be used. In 2004 Daniel Goldston and Cem Yıldırım showed that the constant could be improved further to c = 0.085786… In 2005, Goldston, Pintz and Yıldırım established that c can be chosen arbitrarily small , ; in fact, if one assumes the Elliott-Halberstam conjecture, they show that there are infinitely many n such that at least two of n, n + 2, n + 6, n + 8, n + 12, n + 18, n + 20 are prime.
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.
Defining a Chen prime to be a prime p such that p + 2 is either a prime or a semiprime, Terence Tao and Ben Green showed in 2005 that there are infinitely many three term arithmetic progressions of Chen primes.
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
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" by R. Arenstorf Withdrawn June 8, 2004.