Twin prime conjecture

The twin prime conjecture asserts that infinitely many pairs of prime numbers differ by two, such as 11 and 13. It remains unproved.

Prime number: A natural number greater than one whose only positive divisors are one and itself. Twin primes are pairs drawn from the prime numbers.
Euclid's theorem: The theorem that infinitely many prime numbers exist. It establishes the broader infinitude question from which the twin-prime problem takes its distinctive form.
Yitang Zhang: A Chinese-American mathematician whose 2013 work proved that some bounded gap between primes occurs infinitely often. His breakthrough began the modern bounded-gap program, though its bound was far above two.
Chen's theorem: A theorem stating that infinitely many primes have a prime or semiprime two greater. It comes close to twin primes but allows the second number to have two prime factors.
Twin prime constant: The positive constant in the Hardy–Littlewood prediction for the asymptotic number of twin primes. It quantifies the predicted density of twin primes if the conjectured asymptotic holds.
Sieve theory: A family of methods that estimates or identifies integers by removing those divisible by selected primes. Sieve methods find many prime pairs but struggle to prove infinitely many exact pairs.
Alphonse de Polignac: A French mathematician who proposed in 1849 that every even number occurs infinitely often as a prime gap. His conjecture includes the twin prime conjecture as the case of gap two.
James Maynard: A British mathematician whose sieve methods substantially improved bounds on gaps between primes. His work sharpened bounded-gap results while leaving the twin-prime case unresolved.
Bounded gaps between primes: The theorem that some fixed upper bound on gaps between consecutive primes occurs infinitely often. It proves recurring small gaps but does not show that the gap can be two.
Brun's constant: The convergent sum of the reciprocals of all twin primes. Its finiteness is an unconditional consequence of Brun's theorem, regardless of whether twins are infinite.

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