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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 Ramanujan prime ： ウィキペディア英語版 Ramanujan prime In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function. ==Origins and definition== In 1919, Ramanujan published a new proof of Bertrand's postulate which, as he notes, was first proved by Chebyshev. At the end of the two-page published paper, Ramanujan derived a generalized result, and that is: : $\backslash pi(x)\; -\; \backslash pi(x/2)$ ≥ 1, 2, 3, 4, 5, ... for all ''x'' ≥ 2, 11, 17, 29, 41, ... respectively, where $\backslash pi(x)$ is the prime-counting function, equal to the number of primes less than or equal to ''x''. The converse of this result is the definition of Ramanujan primes: :The ''n''th Ramanujan prime is the least integer ''Rn'' for which $\backslash pi(x)\; -\; \backslash pi(x/2)$ ≥ ''n'', for all ''x'' ≥ ''Rn''. The first five Ramanujan primes are thus 2, 11, 17, 29, and 41. Equivalently, :Ramanujan primes are the least integers ''Rn'' for which there are at least ''n'' primes between ''x'' and ''x''/2 for all ''x'' ≥ ''Rn''. Note that the integer ''Rn'' is necessarily a prime number: $\backslash pi(x)\; -\; \backslash pi(x/2)$ and, hence, $\backslash pi(x)$ must increase by obtaining another prime at ''x'' = ''Rn''. Since $\backslash pi(x)\; -\; \backslash pi(x/2)$ can increase by at most 1, : $\backslash pi($''Rn''$)\; -\; \backslash pi($''Rn''$/2)\; =\; n$. 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「Ramanujan prime」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.