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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 Glaisher's theorem ： ウィキペディア英語版 Glaisher's theorem In number theory, Glaisher's theorem is an identity useful to the study of integer partitions. It is named for James Whitbread Lee Glaisher. It states that the number of partitions of an integer $N$ into parts not divisible by $d$ is equal to the number of partitions of the form :$$ N=N_1+\cdots+N_k where :$N\_i\backslash geq\; N\_$ and :$N\_i\backslash geq\; N\_+1,$ that is, partitions in which no part is repeated ''d'' or more times. When $d=2$ this becomes the special case, known as Euler's theorem, that the number of partitions of $N$ into distinct parts is the same as the number of partitions of $N$ into odd parts. ==Similar theorems== If instead of counting the number of partitions with distinct parts we count the number of partitions with parts differing by at least 2, a theorem similar to Euler's theorem known as Rogers' theorem (after Leonard James Rogers) is obtained: :The number of partitions whose parts differ by at least 2 is equal to the number of partitions involving only numbers congruent to 1 or 4 (mod 5). For example, there are 6 partitions of 10 into parts differing by at least 2, namely 10, 9+1, 8+2, 7+3, 6+4, 6+3+1; and 6 partitions of 10 involving only 1, 4, 6, 9 ..., namely 9+1, 6+4, 6+1+1+1+1, 4+4+1+1, 4+1+1+1+1+1+1, 1+1+1+1+1+1+1+1+1+1. The theorem was discovered independently by Schur and Ramanujan. 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「Glaisher's 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.