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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 Lagrange's four-square theorem ： ウィキペディア英語版 Lagrange's four-square theorem Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as the sum of four integer squares. :$p\; =\; a\_0^2\; +\; a\_1^2\; +\; a\_2^2\; +\; a\_3^2\backslash$ where the four numbers $a\_0,\; a\_1,\; a\_2,\; a\_3$ are integers. For illustration, 3, 31 and 310 can be represented as the sum of four squares as follows: :$$ \begin 3 & = 1^2+1^2+1^2+0^2 \\() 31 & = 5^2+2^2+1^2+1^2 \\() 310 & = 17^2+4^2+2^2+1^2. \end This theorem was proved by Joseph Louis Lagrange in 1770. ==Historical development== From examples given in the ''Arithmetica'' it is clear that Diophantus was aware of the theorem. This book was translated in 1621 into Latin by Bachet (Claude Gaspard Bachet de Méziriac), who stated the theorem in the notes of his translation. But the theorem was not proved until 1770 by Lagrange. Adrien-Marie Legendre completed the theorem in 1797–8 with his three-square theorem, by proving that a positive integer can be expressed as the sum of three squares if and only if it is not of the form $4^k(8m+7)$ for integers $k$ and $m$. Later, in 1834, Carl Gustav Jakob Jacobi discovered a simple formula for the number of representations of an integer as the sum of four squares with his own four-square theorem. The formula is also linked to Descartes' theorem of four "kissing circles", which involves the sum of the squares of the curvatures of four circles. This is also linked to Apollonian gaskets, which were more recently related to the Ramanujan–Petersson conjecture.〔(The Ramanujan Conjecture and some Diophantine Equations ), Peter Sarnak, lecture at Tata Institute of Fundamental Research, of the ICTS lecture series. Bangalore, 2013 via youtube.〕 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「Lagrange's four-square 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.