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Twin prime conjecture

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 that the conjecture is 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.

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, Jing-run Chen showed that there are infinitely many primes p such that p+2 is a product of at most two prime factors. 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, which is concerned with the distribution of twin primes, in analogy to the prime number theorem. Let π2(x) denote the number of primes px such that p + 2 is also prime. Define the twin prime constant C2 as

$C_2 = \prod_{p\ge 3} \frac{p(p-2)}{(p-1)^2} \approx 0.66016118158468695739278121100145 ...$

(here the product extends over all prime numbers p ≥ 3). Then the conjecture is that

$\pi_2(x) \sim 2 C_2 \int_2^x {dt \over (\ln t)^2}$

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.

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