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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 Goldbach–Euler theorem ： ウィキペディア英語版 Goldbach–Euler theorem In mathematics, the Goldbach–Euler theorem (also known as Goldbach's theorem), states that the sum of 1/(''p'' − 1) over the set of perfect powers ''p'', excluding 1 and omitting repetitions, converges to 1: :$\backslash sum\_^\backslash frac=\; +\; \backslash frac\; +\; \backslash frac+\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac+\; \backslash frac\}+\; \backslash cdots\; =\; 1.$ This result was first published in Euler's 1737 paper "''Variæ observationes circa series infinitas''". Euler attributed the result to a letter (now lost) from Goldbach. ==Proof== Goldbach's original proof to Euler involved assigning a constant to the harmonic series: $\backslash textstyle\; x\; =\; \backslash sum\_^\backslash infty\; \backslash frac\backslash$, which is divergent. Such a proof is not considered rigorous by modern standards. It is also interesting to note that there is a strong resemblance between the method of sieving out powers employed in his proof and the method of factorization used to derive Euler's product formula for the Riemann zeta function. Let x be given by :$x\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; \backslash cdots$ Since the sum of the reciprocal of every power of two is $\backslash textstyle\; 1\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$, subtracting the terms with powers of two from x gives :$x\; -\; 1\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$ Repeat the process with the terms with the powers of three: $\backslash textstyle\; \backslash frac\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$ :$x\; -\; 1\; -\; \backslash frac\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$ Absent from the above sum are now all terms with powers of two and three. Continue by removing terms with powers of 5, 6 and so on until the right side is exhausted to the value of 1. Eventually, we obtain the equation :$x\; -\; 1\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash frac\; -\; \backslash cdots\; =\; 1$ which we rearrange into :$x\; -\; 1\; =\; 1\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac\; +\; \backslash cdots$ where the denominators consist of all positive integers that are the non-powers minus one. By subtracting the previous equation from the definition of x given above, we obtain :$1\; =\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac+\; \backslash frac\; +\; \backslash frac\; +\; \backslash frac+\; \backslash frac\; +\; \backslash cdots$ where the denominators now consist only of perfect powers minus one. While mathematically incorrect, Goldbach's proof provides a reasonably intuitive visualization of the problem. Rigorous proofs require proper and more careful treatment of the divergent terms of the harmonic series. Other proofs make use of the fact that the sum of 1/''p'' over the set of perfect powers ''p'', excluding 1 but including repetitions, converges to 1 by demonstrating the equivalence: :$\backslash sum\_^\backslash frac\; =\; \backslash sum\_^\backslash infty\; \backslash sum\_^\backslash infty\; \backslash frac\; =\; 1.$ 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「Goldbach–Euler 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.