Twin prime conjecture

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The twin prime conjecture is a famous unsolved problem in number theory that involves prime numbers. It states:

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. The majority of mathematicians believe the conjecture to be true, based 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 which have a distance of 2k. The case k = 1 is the twin prime conjecture.

Table of contents

1 Partial results
2 Hardy-Littlewood conjecture
3 See also

Partial results

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 a 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.

Hardy-Littlewood conjecture

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.