Twin Prime

Twin primes are pairs of prime numbers that differ from each other by exactly two, such as three and five, eleven and thirteen, or seventeen and nineteen, making them the closest two odd prime numbers can ever be to one another. The twin prime conjecture, one of the oldest unsolved problems in number theory, proposes that infinitely many such pairs exist, a claim still unproved despite being checked computationally for enormous ranges of numbers where the pairs keep appearing. A major advance came in 2013, when the mathematician Yitang Zhang proved that there are infinitely many pairs of primes differing by some bounded gap, not necessarily exactly two, an unconditional result that had eluded mathematicians for decades; the Polymath Project and the mathematician James Maynard subsequently lowered Zhang's original bound substantially using different and more flexible methods. Twin primes become rarer, on average, as numbers grow larger, a thinning-out pattern described qualitatively by an estimate known as the Hardy-Littlewood conjecture, though no result has ever matched this qualitative estimate to the still-unproven claim that infinitely many twin primes exist.