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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 De Polignac's formula ： ウィキペディア英語版 De Polignac's formula In number theory, de Polignac's formula, named after Alphonse de Polignac, gives the prime decomposition of the factorial ''n''!, where ''n'' ≥ 1 is an integer. L. E. Dickson attributes the formula to Legendre.〔Leonard Eugene Dickson, ''History of the Theory of Numbers'', Volume 1, Carnegie Institution of Washington, 1919, page 263.〕 ==The formula== Let ''n'' ≥ 1 be an integer. The prime decomposition of ''n''! is given by :$n!\; =\; \backslash prod\_\; p^,$ where :$s\_p(n)\; =\; \backslash sum\_^\backslash infty\; \backslash left\backslash lfloor\backslash frac\backslash right\backslash rfloor,$ and the brackets represent the floor function. Note that the former product can equally well be taken only over primes less than or equal to ''n'', and the latter sum can equally well be taken for ''j'' ranging from ''1'' to log''p''(''n''), i.e : :$s\_p(n)\; =\; \backslash sum\_^\; \backslash left\backslash lfloor\backslash frac\backslash right\backslash rfloor$ Note that, for any real number ''x'', and any integer ''n'', we have: :$\backslash left\backslash lfloor\backslash frac\backslash right\backslash rfloor\; =\; \backslash left\backslash lfloor\backslash frac\backslash right\backslash rfloor$ which allows one to more easily compute the terms ''s''''p''(''n''). The small disadvantage of the De Polignac's formula is that we need to know all the primes up to ''n''. In fact, :$\backslash displaystyle\; n!\; =\; \backslash prod\_^\; p\_^\; =\; \backslash prod\_^\; p\_i^(n)\; \backslash rfloor\}\; \backslash left\backslash lfloor\backslash frac^j\}\backslash right\backslash rfloor\; \}$ where $\backslash pi(n)$ is a prime-counting function counting the number of prime numbers less than or equal to ''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「De Polignac's formula」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.