Twin prime conjecture
The twin prime conjecture is a famous unsolved problem in number theory that involves prime numbers. Its weak form 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 Alphonse 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, Paul Erdös showed that there is a constant k and infinitely many primes p such that p' - p < k ln(p), where p' denotes the next prime after p.
In 1966, Jing-run Chen showed that there are an infinite number of 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 strong form of this conjecture, the so called 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 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
- π2(x) ~ 2 C2 ∫2x dt / (ln t)2
in the sense that the limit of the quotient of the two expressions is 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.
See also: twin prime, Brun's constant.