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twin prime
A twin prime is a prime number that has a prime gap of two. In other words, to qualify as a twin prime, the prime number must be either 2 less or 2 more than another prime number (which by definition would mean that it, too, is a twin prime)—for example, the twin prime pair (41, 43). Two is not considered a twin prime with the number three, since it violates the aforementioned rule.〔https://primes.utm.edu/lists/small/100ktwins.txt〕 Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes appear despite the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger due to the prime number theorem (the "average gap" between primes less than ''n'' is log(''n'')).
==History==
The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states: ''There are infinitely many primes'' ''p'' ''such that'' ''p'' + 2 ''is also prime.'' In 1849, de Polignac made the more general conjecture that for every natural number ''k'', there are infinitely many prime pairs ''p'' and ''p''′ such that ''p''′ − ''p'' = 2''k''. The case ''k'' = 1 is the twin prime conjecture.
A stronger form of the twin prime conjecture, the Hardy–Littlewood conjecture (see below), postulates a distribution law for twin primes akin to the prime number theorem.
On April 17, 2013, Yitang Zhang announced a proof that for some integer ''N'' that is less than 70 million, there are infinitely many pairs of primes that differ by ''N''. Zhang's paper was accepted by ''Annals of Mathematics'' in early May 2013. Terence Tao subsequently proposed a Polymath Project collaborative effort to optimize Zhang’s bound. As of April 14, 2014, one year after Zhang's announcement, according to the Polymath project wiki, the bound has been reduced to 246.〔(【引用サイトリンク】title=Bounded gaps between primes )〕 Further, assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that the bound has been reduced to 12 and 6, respectively.〔(【引用サイトリンク】title=Bounded gaps between primes )〕 These improved bounds were discovered using a different approach that was simpler than Zhang's and was discovered independently by James Maynard and Terence Tao. This second approach also gave bounds for the smallest ''f''(''m'') needed to guarantee that infinitely many intervals of width ''f''(''m'') contain at least ''m'' primes.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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