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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 Green–Tao theorem ： ウィキペディア英語版 Green–Tao theorem In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004,〔.〕 states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number ''k'', no matter how big, there exist arithmetic progressions of primes with ''k'' terms. The proof is an extension of Szemerédi's theorem. ==Statement== Let $\backslash pi(N)$ denote the number of primes less than or equal to $N$. If $A$ is a subset of the prime numbers such that : $\backslash limsup\_\; \backslash dfrac>0$, then for all positive integers $k$, the set $A$ contains infinitely many arithmetic progressions of length $k$. In particular, the entire set of prime numbers contains arbitrarily long arithmetic progressions. In their later work on the generalized Hardy–Littlewood conjecture, Green and Tao derived the asymptotic formula : $(\backslash mathfrak\_k\; +\; o(1))\backslash frac$ for the number of ''k'' tuples of primes in arithmetic progression. Here, $\backslash mathfrak\_k$ is the constant : $\backslash mathfrak\_k\; :=\; \backslash frac1\backslash left(\backslash prod\_\backslash fracp\backslash left(\backslash frac\backslash right)^\backslash right)\backslash left(\backslash prod\_\backslash left(1\; -\; \backslash fracp\backslash right)\backslash left(\backslash frac\backslash right)^\backslash right)$. 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「Green–Tao theorem」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.