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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 Euclid's theorem ： ウィキペディア英語版 Euclid's theorem Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. There are several well-known proofs of the theorem. ==Euclid's proof== Euclid offered the following proof published in his work ''Elements'' (Book IX, Proposition 20),〔James Williamson (translator and commentator), ''The Elements of Euclid, With Dissertations'', Clarendon Press, Oxford, 1782, page 63.〕 which is paraphrased here. Consider any finite list of prime numbers ''p''1, ''p''2, ..., ''p''''n''. It will be shown that at least one additional prime number not in this list exists. Let ''P'' be the product of all the prime numbers in the list: ''P'' = ''p''1''p''2...''p''''n''. Let ''q'' = ''P'' + 1. Then ''q'' is either prime or not: *If ''q'' is prime, then there is at least one more prime than is in the list. *If ''q'' is not prime, then some prime factor ''p'' divides ''q''. If this factor ''p'' were on our list, then it would divide ''P'' (since ''P'' is the product of every number on the list); but ''p'' divides ''P'' + 1 = ''q''. If ''p'' divides ''P'' and ''q,'' then ''p'' would have to divide the difference〔In general, for any integers ''a'', ''b'', ''c'' if $a\; \backslash mid\; b$ and $a\; \backslash mid\; c$, then $a\; \backslash mid\; (b\; -\; c)$. For more information, see Divisibility.〕 of the two numbers, which is (''P'' + 1) − ''P'' or just 1. Since no prime number divides 1, this would be a contradiction and so ''p'' cannot be on the list. This means that at least one more prime number exists beyond those in the list. This proves that for every finite list of prime numbers there is a prime number not on the list, and therefore there must be infinitely many prime numbers. Euclid is often erroneously reported to have proved this result by contradiction, beginning with the assumption that the set initially considered contains all prime numbers, or that it contains precisely the ''n'' smallest primes, rather than any arbitrary finite set of primes.〔Michael Hardy and Catherine Woodgold, "Prime Simplicity", ''Mathematical Intelligencer'', volume 31, number 4, fall 2009, pages 44–52.〕 Although the proof as a whole is not by contradiction (it does not assume that only finitely many primes exist), a proof by contradiction is within it, which is that none of the initially considered primes can divide the number ''q'' above. 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「Euclid'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.