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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 Singmaster's conjecture ： ウィキペディア英語版 Singmaster's conjecture Singmaster's conjecture is a conjecture in combinatorial number theory in mathematics, named after the British mathematician David Singmaster who proposed it in 1971. It says that there is a finite upper bound on the multiplicities of entries in Pascal's triangle (other than the number 1, which appears infinitely many times). It is clear that the only number that appears infinitely many times in Pascal's triangle is 1, because any other number ''x'' can appear only within the first ''x'' + 1 rows of the triangle. Paul Erdős said that Singmaster's conjecture is probably true but he suspected it would be very hard to prove. Let ''N''(''a'') be the number of times the number ''a'' > 1 appears in Pascal's triangle. In big O notation, the conjecture is: :$N(a)\; =\; O(1).\backslash ,$ ==Known results== Singmaster (1971) showed that :$N(a)\; =\; O(\backslash log\; a).\backslash ,$ Abbot, Erdős, and Hanson (see References) refined the estimate. The best currently known (unconditional) bound is :$N(a)\; =\; O\backslash left(\backslash frac\backslash right),\backslash ,$ and is due to Kane (2007). Abbot, Erdős, and Hanson note that conditional on Cramér's conjecture on gaps between consecutive primes that :$N(a)\; =\; O\backslash left(\; \backslash log(a)^\backslash right)$ holds for every $\backslash varepsilon\; >\; 0$. Singmaster (1975) showed that the Diophantine equation :$=\; ,$ has infinitely many solutions for the two variables ''n'', ''k''. It follows that there are infinitely many entries of multiplicity at least 6. The solutions are given by :$n\; =\; F\_\; F\_\; -\; 1,\backslash ,$ :$k\; =\; F\_\; F\_\; -\; 1,\backslash ,$ where ''F''''n'' is the ''n''th Fibonacci number (indexed according to the convention that ''F''1 = ''F''2 = 1). 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「Singmaster's conjecture」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.