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In number theory, a formula for primes is a formula generating the prime numbers, exactly and without exception. No such formula which is efficiently computable is known. A number of constraints are known, showing what such a "formula" can and cannot be. ==Prime formulas and polynomial functions== It is known that no non-constant polynomial function ''P''(''n'') with integer coefficients exists that evaluates to a prime number for all integers ''n''. The proof is as follows: Suppose such a polynomial existed. Then ''P''(1) would evaluate to a prime ''p'', so . But for any ''k'', also, so cannot also be prime (as it would be divisible by ''p'') unless it were ''p'' itself, but the only way for all ''k'' is if the polynomial function is constant. The same reasoning shows an even stronger result: no non-constant polynomial function ''P''(''n'') exists that evaluates to a prime number for almost all integers ''n''. Euler first noticed (in 1772) that the quadratic polynomial :''P''(''n'') = ''n''2 + ''n'' + 41 is prime for all natural numbers less than 40. The primes for ''n'' = 0, 1, 2, ... are 41, 43, 47, 53, 61, 71... The differences between the terms are 2, 4, 6, 8, 10... For ''n'' = 40, it produces a square number, 1681, which is equal to 41×41, the smallest composite number for this formula. If 41 divides ''n'', it divides ''P(n)'' too. The phenomenon is related to the Ulam spiral, which is also implicitly quadratic, and the class number; this polynomial is related to the Heegner number , and there are analogous polynomials for (the lucky numbers of Euler), corresponding to other Heegner numbers. It is known, based on Dirichlet's theorem on arithmetic progressions, that linear polynomial functions produce infinitely many primes as long as ''a'' and ''b'' are relatively prime (though no such function will assume prime values for all values of ''n''). Moreover, the Green–Tao theorem says that for any ''k'' there exists a pair of ''a'' and ''b'' with the property that is prime for any ''n'' from 0 to ''k'' − 1. However, the best known result of such type is for ''k'' = 26 (by Benoãt Perichon of France): :43142746595714191 + 5283234035979900''n'' is prime for all ''n'' from 0 to 25 . It is not even known whether there exists a univariate polynomial of degree at least 2 that assumes an infinite number of values that are prime; see Bunyakovsky conjecture. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Formula for primes」の詳細全文を読む スポンサード リンク
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