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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 Aurifeuillean factorization ： ウィキペディア英語版 Aurifeuillean factorization In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is a special type of algebraic factorization that comes from non-trivial factorizations of cyclotomic polynomials over the integers. Although cyclotomic polynomials themselves are irreducible over the integers, when restricted to particular integer values they may have an algebraic factorization, as in the examples below. == Examples == * Numbers of the form $2^+1$ have the following aurifeuillean factorization: :: $2^+1\; =\; (2^-2^+1)\backslash cdot\; (2^+2^+1)$ * Numbers of the form $b^n\; -\; 1$ or $\backslash Phi\_n(b)$, where $b\; =\; s^2\backslash cdot\; t$ with square-free $t$, have aurifeuillean factorization if and only if one of the following conditions holds: * * $t\backslash equiv\; 1\; \backslash pmod\; 4$ and $n\backslash equiv\; t\; \backslash pmod$ * * $t\backslash equiv\; 2,\; 3\; \backslash pmod\; 4$ and $n\backslash equiv\; 2t\; \backslash pmod$ : Thus, when $b\; =\; s^2\backslash cdot\; t$ with square-free $t$, and $n$ is congruent to $t$ mod $2t$, then if $t$ is congruent to 1 mod 4, $b^n-1$ have aurifeuillean factorization, otherwise, $b^n+1$ have aurifeuillean factorization. : When the number is of a particular form (the exact expression varies with the base), Aurifeuillian factorization may be used, which gives a product of two or three numbers. The following equations give Aurifeuillian factors for the Cunningham project bases as a product of ''F'', ''L'' and ''M'':〔(【引用サイトリンク】url=http://homes.cerias.purdue.edu/~ssw/cun/pmain1211 ) At the end of tables 2LM, 3+, 5-, 7+, 10+, 11+ and 12+ are formulae detailing the Aurifeuillian factorisations.〕 : If we let ''L'' = ''A'' - ''B'', ''M'' = ''A'' + ''B'', the Aurifeuillian factorizations for ''b''''n'' ± 1 with the base 2 ≤ ''b'' ≤ 24 (perfect powers excluded, since a power of ''b''''n'' is also a power of ''b'') are: : (Number = ''F'' * (''A'' - ''B'') * (''A'' + ''B'') = ''F'' * ''L'' * ''M'') : : (See 〔(List of Aurifeuillean factorization )〕 for more information (square-free bases up to 199)) : In fact, ''L'' * ''M'' = $\backslash Phi\_n(b^)$, where $b\; =\; s^2\backslash cdot\; t$ with square-free $t$, and ''n'' = ''t'' if ''t'' is congruent to 1 mod 4. Otherwise, ''n'' = 2''t''. * Numbers of the form $a^4\; +\; 4b^4$ have the following aurifeuillean factorization: :: $a^4\; +\; 4b^4\; =\; (a^2\; -\; 2ab\; +\; 2b^2)\backslash cdot\; (a^2\; +\; 2ab\; +\; 2b^2).$ * Lucas number $L\_$ have the following aurifeuillean factorization: :: $L\_\; =\; L\_\backslash cdot\; (5+1)\backslash cdot\; (5+1)$ : where $L\_n$ is the $n$th Lucas number, $F\_n$ is the $n$th Fibonacci number. 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「Aurifeuillean factorization」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.