twinprime

In number theory, a twin prime is a pair of prime numbers that differ by 2. If p is prime and p+2 is also prime, then (p, p+2) is a twin prime pair. The smaller member is typically called the twin prime, and the larger its mate.

Examples of twin primes include (3,5), (5,7), (11,13), (17,19), (29,31), and (41,43). Note that twin prime pairs can share a member, as in the pairs (3,5) and (5,7); both are valid twin prime pairs.

The central question about twin primes is whether there are infinitely many of them. This is known as the Twin Prime Conjecture. A widely cited heuristic, due to Hardy and Littlewood, predicts an asymptotic formula for the number of twin primes up to x. If pi2(x) denotes the number of twin prime pairs with smaller member at most x, then pi2(x) is conjectured to be approximately 2 C2 ∫_2^x dt/(log t)^2, where C2 is the twin prime constant, about 0.6601618. In simpler terms, pi2(x) is expected to grow roughly like 2 C2 x/(log x)^2 as x grows large.

As of recent years, the infinitude of twin primes remains unproven. Substantial progress has been made in related areas of prime gaps, and numerical computations have verified the existence of many twin primes up to very large bounds, providing strong evidence in favor of the conjecture. Twin primes occupy a central place in analytic number theory and are connected to broader prime-k-tuple conjectures and sieve methods.

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