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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 Cramér's conjecture ： ウィキペディア英語版 Cramér's conjecture In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that :$p\_-p\_n=O((\backslash log\; p\_n)^2),\backslash$ where ''p''''n'' denotes the ''n''th prime number, ''O'' is big O notation, and "log" is the natural logarithm. While this is the statement explicitly conjectured by Cramér, his argument actually supports the stronger statement :$\backslash limsup\_\; \backslash frac\; =\; 1,$ and this formulation is often called Cramér's conjecture in the literature. Neither form of Cramér's conjecture has yet been proven or disproven. ==Conditional proven results on prime gaps== Cramér gave a conditional proof of the much weaker statement that :$p\_-p\_n\; =\; O(\backslash sqrt\backslash ,\backslash log\; p\_n)$ on the assumption of the Riemann hypothesis.〔 In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,〔.〕 :$\backslash limsup\_\backslash frac=\backslash infty.$ 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「Cramér's 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.