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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Bateman–Horn conjecture ： ウィキペディア英語版 Bateman–Horn conjecture In number theory, the Bateman–Horn conjecture is a statement concerning the frequency of prime numbers among the values of a system of polynomials, named after mathematicians Paul T. Bateman and Roger A Horn, of The University of Utah, who proposed it in 1962. It provides a vast generalization of such conjectures as the Hardy and Littlewood conjecture on the density of twin primes or their conjecture on primes of the form ''n''2 + 1; it is also a strengthening of Schinzel's hypothesis H. It remains unsolved as of January 2014. ==Definition== The Bateman–Horn conjecture provides a conjectured density for the positive integers at which a given set of polynomials all have prime values. For a set of ''m'' distinct irreducible polynomials ''ƒ''1, ..., ''ƒ''''m'' with integer coefficients, an obvious necessary condition for the polynomials to simultaneously generate prime values infinitely often is that they satisfy Bunyakovsky's property, that there does not exist a prime number ''p'' that divides their product ''f''(''n'') for every positive integer ''n''. For, if not, then one of the values of the polynomials must be equal to ''p'', which can only happen for finitely many values of ''n''. An integer ''n'' is prime-generating for the given system of polynomials if every polynomial ''ƒi''(''n'') produces a prime number when given ''n'' as its argument. If ''P(x)'' is the number of prime-generating integers among the positive integers less than ''x'', then the Bateman–Horn conjecture states that :$P(x)\; \backslash sim\; \backslash frac\; \backslash int\_2^x\; \backslash frac,\backslash ,$ where ''D'' is the product of the degrees of the polynomials and where ''C'' is the product over primes ''p'' :$C\; =\; \backslash prod\_p\; \backslash frac\backslash$ with $N(p)$ the number of solutions to :$f(n)\; \backslash equiv\; 0\; \backslash pmod\; p.\backslash$ Bunyakovsky's property implies for all primes ''p'', so each factor in the infinite product ''C'' is positive. Intuitively one then naturally expects that the constant ''C'' is itself positive, and with some work this can be proved. (Work is needed since some infinite products of positive numbers equal zero.) 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「Bateman–Horn conjecture」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.