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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 Sums of powers ： ウィキペディア英語版 Sums of powers In mathematics and statistics, sums of powers occur in a number of contexts: *Sums of squares arise in many contexts. *Faulhaber's formula expresses $1^k\; +\; 2^k\; +\; 3^k\; +\; \backslash cdots\; +\; n^k$ as a polynomial in ''n''. *Fermat's right triangle theorem states that there is no solution in positive integers for $a^4=b^4+c^2.$ *Fermat's Last Theorem states that $x^k+y^k=z^k$ is impossible in positive integers with ''k''>2. *The equation of a superellipse is $|x/a|^k+|y/b|^k=1$. The squircle is the case $k=4,\; a=b$. *Euler's sum of powers conjecture (disproved) concerns situations in which the sum of ''n'' integers, each a ''k''th power of an integer, equals another ''k''th power. *The Fermat-Catalan conjecture asks whether there are an infinitude of examples in which the sum of two coprime integers, each a power of an integer, with the powers not necessarily equal, can equal another integer that is a power, with the reciprocals of the three powers summing to less than 1. *Beal's conjecture concerns the question of whether the sum of two coprime integers, each a power greater than 2 of an integer, with the powers not necessarily equal, can equal another integer that is a power greater than 2. *The Jacobi–Madden equation is $a^4\; +\; b^4\; +\; c^4\; +\; d^4\; =\; (a\; +\; b\; +\; c\; +\; d)^4$ in integers. *The Prouhet–Tarry–Escott problem considers sums of two sets of ''k''th powers of integers that are equal for multiple values of ''k''. *A taxicab number is the smallest integer that can be expressed as a sum of two positive third powers in ''n'' distinct ways. *The Riemann zeta function is the sum of the reciprocals of the positive integers each raised to the power ''s'', where ''s'' is a complex number whose real part is greater than 1. *The Lander, Parkin, and Selfridge conjecture concerns the minimal value of ''m'' + ''n'' in $\backslash sum\_^\; a\_i^k\; =\; \backslash sum\_^\; b\_j^k.$ *Waring's problem asks whether for every natural number k there exists an associated positive integer ''s'' such that every natural number is the sum of at most ''s k''th powers of natural numbers. *The successive powers of the golden ratio ''φ'' obey the Fibonacci recurrence: ::$\backslash varphi^$ = \varphi^n + \varphi^. *Newton's identities express the sum of the ''k''th powers of all the roots of a polynomial in terms of the coefficients in the polynomial. *The sum of cubes of numbers in arithmetic progression is sometimes another cube. *The Fermat cubic, in which the sum of three cubes equals another cube, has a general solution. *The power sum symmetric polynomial is a building block for symmetric polynomials. *The sum of the reciprocals of all perfect powers including duplicates (but not including 1) equals 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「Sums of powers」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.