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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 Fermat quotient ： ウィキペディア英語版 Fermat quotient In number theory, the Fermat quotient of an integer ''a'' with respect to an odd prime ''p'' is defined as:〔(Fermat Quotient ) at ''The Prime Glossary''〕〔Paulo Ribenboim, ''13 Lectures on Fermat's Last Theorem'' (1979), especially pp. 152, 159-161.〕〔Paulo Ribenboim, ''My Numbers, My Friends: Popular Lectures on Number Theory'' (2000), p. 216.〕 :$q\_p(a)\; =\; \backslash frac.$ or :$\backslash delta\_p(a)\; =\; \backslash frac$. This article is about the former. For the latter see p-derivation. If the base ''a'' is coprime to the exponent ''p'' then Fermat's little theorem says that ''q''''p''(''a'') will be an integer. The quotient is named after Pierre de Fermat. ==Properties== From the definition, it is obvious that :$q\_p(1)\; \backslash equiv\; 0\; \backslash pmod$ :$q\_p(-a)\; \backslash equiv\; q\_p(a)\; \backslash pmod$, since ''p'' − 1 is even. In 1850 Gotthold Eisenstein proved that if ''a'' and ''b'' are both coprime to ''p'', then:〔Gotthold Eisenstein, "Neue Gattung zahlentheoret. Funktionen, die v. 2 Elementen abhangen und durch gewisse lineare Funktional-Gleichungen definirt werden," ''Bericht über die zur Bekanntmachung geeigneten Verhandlungen der Königl. Preuß. Akademie der Wissenschaften zu Berlin'' 1850, 36-42〕 :$q\_p(ab)\backslash equiv\; q\_p(a)+q\_p(b)\; \backslash pmod$; :$q\_p(a^r)\backslash equiv\; rq\_p(a)\; \backslash pmod$; :$q\_p(p-a)\backslash equiv\; q\_p(a)\; +\; \backslash frac\; \backslash pmod$; :$q\_p(p+a)\backslash equiv\; q\_p(a)\; -\; \backslash frac\; \backslash pmod$; :$q\_p(p-1)\backslash equiv\; 1\; \backslash pmod$; :$q\_p(p+1)\backslash equiv\; -1\; \backslash pmod$. Eisenstein likened the first two of these congruences to properties of logarithms. These properties imply :$q\_p(1/a)\; \backslash equiv\; -q\_p(a)\; \backslash pmod$; :$q\_p(a/b)\; \backslash equiv\; q\_p(a)\; -\; q\_p(b)\; \backslash pmod$. In 1895 Dmitry Mirimanoff pointed out that an iteration of Eisenstein's rules gives the corollary:〔Dmitry Mirimanoff, "Sur la congruence (''r''''p'' − 1 − 1):''p'' = ''q''r'''' (mod ''p'')," ''Journal für die reine und angewandte Mathematik'' 115 (1895): 295-300〕 :$q\_p(a+np)\backslash equiv\; q\_p(a)-n\backslash cdot\backslash frac\; \backslash pmod.$ From this, it follows〔Paul Bachmann, ''Niedere Zahlentheorie'', 2 vols. (Leipzig, 1902), 1:159.〕 that :$q\_p(a+np^2)\backslash equiv\; q\_p(a)\; \backslash pmod.$ 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「Fermat quotient」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.