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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 Bertrand's postulate ： ウィキペディア英語版 Bertrand's postulate Bertrand's postulate is a theorem stating that for any integer $n\; >\; 3$, there always exists at least one prime number $p$ with :. A weaker but more elegant formulation is: for every $n\; >\; 1$ there is always at least one prime $p$ such that :. Another formulation, where $p\_n$ is the $n$-th prime, is for $n\; \backslash ge\; 1$ :. This statement was first conjectured in 1845 by Joseph Bertrand 〔Joseph Bertrand. Mémoire sur le nombre de valeurs que peut prendre une fonction quand on y permute les lettres qu'elle renferme. Journal de l'Ecole Royale Polytechnique, Cahier 30, Vol. 18 (1845), 123-140.〕 (1822–1900). Bertrand himself verified his statement for all numbers in the interval His conjecture was completely proved by Chebyshev (1821–1894) in 1852〔P. Tchebychev. Mémoire sur les nombres premiers. Journal de mathématiques pures et appliquées, Sér. 1(1852), 366-390. (Proof of the postulate: 371-382). Also see Mémoires de l'Académie Impériale des Sciences de St. Pétersbourg, vol. 7, pp.15-33, 1854〕 and so the postulate is also called the Bertrand–Chebyshev theorem or Chebyshev's theorem. Chebyshev's theorem can also be stated as a relationship with $\backslash scriptstyle\; \backslash pi(x)\; \backslash ,$, where $\backslash scriptstyle\; \backslash pi(x)\; \backslash ,$ is the prime counting function (number of primes less than or equal to $\backslash scriptstyle\; x\; \backslash ,$): :$\backslash pi(x)\; -\; \backslash pi(\backslash tfrac)\; \backslash ge\; 1,\backslash ,$ for all $\backslash ,\; x\; \backslash ge\; 2.\; \backslash ,$ In 1919, Ramanujan (1887–1920) used properties of the Gamma function to give a simpler proof, from which the concept of Ramanujan primes would later arise, and Erdős (1913–1996) in 1932 published a simpler proof using binomial coefficients and the Chebyshev function ''ϑ'', defined as: : $\backslash vartheta(x)\; =\; \backslash sum\_^\; \backslash ln\; (p)$ where ''p'' ≤ ''x'' runs over primes. See proof of Bertrand's postulate for the details. == Sylvester's theorem == Bertrand's postulate was proposed for applications to permutation groups. Sylvester (1814–1897) generalized the weaker statement with the statement: the product of ''k'' consecutive integers greater than ''k'' is divisible by a prime greater than ''k''. Bertrand's (weaker) postulate follows from this by taking ''k = n'', and considering the ''k'' numbers ''n''+1, ''n''+2, up to and including ''n''+''k'' = 2''n'', where ''n'' > 1. According to Sylvester's generalization, one of these numbers has a prime factor greater than ''k''. Since all these numbers are less than 2(''k''+1), the number with a prime factor greater than ''k'' has only one prime factor, and thus is a prime. Note that 2''n'' is not prime, and thus indeed we now know there exists a prime ''p'' with ''n'' < ''p'' < 2''n''. 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「Bertrand's postulate」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.