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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 Wolstenholme's theorem ： ウィキペディア英語版 Wolstenholme's theorem In mathematics, Wolstenholme's theorem states that for a prime number ''p'' > 3, the congruence :$\backslash equiv\; 1\; \backslash pmod$ holds, where the parentheses denote a binomial coefficient. For example, with ''p'' = 7, this says that 1716 is one more than a multiple of 343. An equivalent formulation is the congruence :$\backslash equiv\; \backslash pmod.$ The theorem was first proved by Joseph Wolstenholme in 1862. In 1819, Charles Babbage showed the same congruence modulo ''p''2, which holds for all primes ''p'' (for ''p''=2 only in the second formulation). The second formulation of Wolstenholme's theorem is due to J. W. L. Glaisher and is inspired by Lucas' theorem. No known composite numbers satisfy Wolstenholme's theorem and it is conjectured that there are none (see below). A prime that satisfies the congruence modulo ''p''4 is called a Wolstenholme prime (see below). As Wolstenholme himself established, his theorem can also be expressed as a pair of congruences for (generalized) harmonic numbers: :$1+++...+\; \backslash equiv\; 0\; \backslash pmod\; \backslash mbox$ :$1+++...+\; \backslash equiv\; 0\; \backslash pmod\; p.$ (Congruences with fractions make sense, provided that the denominators are coprime to the modulus.) For example, with ''p''=7, the first of these says that the numerator of 49/20 is a multiple of 49, while the second says the numerator of 5369/3600 is a multiple of 7. == Wolstenholme primes == (詳細はiff the following condition holds: :$\backslash equiv\; 1\; \backslash pmod.$ If ''p'' is a Wolstenholme prime, then Glaisher's theorem holds modulo ''p''4. The only known Wolstenholme primes so far are 16843 and 2124679 ; any other Wolstenholme prime must be greater than 109. This result is consistent with the heuristic argument that the residue modulo ''p''4 is a pseudo-random multiple of ''p''3. This heuristic predicts that the number of Wolstenholme primes between ''K'' and ''N'' is roughly ''ln ln N − ln ln K''. The Wolstenholme condition has been checked up to 109, and the heuristic says that there should be roughly one Wolstenholme prime between 109 and 1024. A similar heuristic predicts that there are no "doubly Wolstenholme" primes, for which the congruence would hold modulo ''p''5. 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「Wolstenholme'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.