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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 perfect power ： ウィキペディア英語版 perfect power In mathematics, a perfect power is a positive integer that can be expressed as an integer power of another positive integer. More formally, ''n'' is a perfect power if there exist natural numbers ''m'' > 1, and ''k'' > 1 such that ''mk'' = ''n''. In this case, ''n'' may be called a perfect ''k''th power. If ''k'' = 2 or ''k'' = 3, then ''n'' is called a perfect square or perfect cube, respectively. Sometimes 1 is also considered a perfect power (1''k'' = 1 for any ''k''). == Examples and sums == A sequence of perfect powers can be generated by iterating through the possible values for ''m'' and ''k''. The first few ascending perfect powers in numerical order (showing duplicate powers) are : :$2^2\; =\; 4,\backslash \; 2^3\; =\; 8,\backslash \; 3^2\; =\; 9,\backslash \; 2^4\; =\; 16,\backslash \; 4^2\; =\; 16,\backslash \; 5^2\; =\; 25,\backslash \; 3^3\; =\; 27,\backslash$ $2^5\; =\; 32,\backslash \; 6^2\; =\; 36,\backslash \; 7^2\; =\; 49,\backslash \; 2^6\; =\; 64,\backslash \; 4^3\; =\; 64,\backslash \; 8^2\; =\; 64,\; \backslash dots$ The sum of the reciprocals of the perfect powers (including duplicates) is 1: :$\backslash sum\_^\; \backslash sum\_^\backslash frac=1.$ which can be proved as follows: :$\backslash sum\_^\; \backslash sum\_^\backslash frac$ =\sum_^ \frac \sum_^\frac =\sum_^ \frac \left( \frac \right) =\sum_^ \frac =\sum_^ \left( \frac - \frac \right) = 1 \, . The first perfect powers without duplicates are (): :(sometimes 1), 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361, 400, 441, 484, ... The sum of the reciprocals of the perfect powers ''p'' without duplicates is: :$\backslash sum\_\backslash frac=\backslash sum\_^\backslash mu(k)(1-\backslash zeta(k))\; \backslash approx\; 0.874464368\; \backslash dots$ where μ(''k'') is the Möbius function and ζ(''k'') is the Riemann zeta function. According to Euler, Goldbach showed (in a now lost letter) that the sum of 1/(''p''−1) over the set of perfect powers ''p'', excluding 1 and excluding duplicates, is 1: :$\backslash sum\_\backslash frac=\; +\; \backslash frac\; +\; \backslash frac+\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac+\; \backslash frac\}+\; \backslash cdots\; =\; 1.$ This is sometimes known as the Goldbach-Euler theorem. 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「perfect power」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.