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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 Büchi's problem ： ウィキペディア英語版 Büchi's problem Büchi's problem, also known as the ''n'' squares' problem, is an open problem from number theory named after the Swiss mathematician Julius Richard Büchi. It asks whether there is a positive integer ''M'' such that every sequence of ''M'' or more integer squares, whose second difference is constant and equal to 2, is necessarily a sequence of squares of the form (''x'' + ''i'')2, ''i'' = 1, 2, ..., ''M'',... for some integer ''x''. In 1983, Douglas Hensley observed that Büchi's problem is equivalent to the following: Does there exist a positive integer ''M'' such that, for all integers ''x'' and ''a'', the quantity (''x'' + ''n'')2 + ''a'' cannot be a square for more than ''M'' consecutive values of ''n'', unless ''a'' = 0? ==Statement of Büchi's problem== Büchi's problem can be stated in the following way: Does there exist a positive integer ''M'' such that the system of equations : $$ \begin x_2^2-2x_1^2+x_0^2=2\\ x_3^2-2x_2^2+x_1^2=2\\ ^2+x_^2=2 \end has only solutions satisfying $x\_n^2=(x\_0+n)^2.$ Since the first difference of the sequence $\backslash sigma=(x\_n^2)\_$ is the sequence $\backslash Delta^(\backslash sigma)=(x\_^2-x\_n^2)\_$, the second difference of $\backslash sigma$ is : $$ \Delta^(\sigma)=((x_^2-x_^2)-(x_^2-x_n^2))_=(x_^2-2x_^2+x_n^2)_. Therefore, the above system of equations is equivalent to the single equation : $\backslash Delta^(\backslash sigma)=(2)\_$ where the unknown is the sequence $\backslash sigma$. 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「Büchi's problem」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.