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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 Odd greedy expansion ： ウィキペディア英語版 Odd greedy expansion In number theory, the odd greedy expansion problem concerns a method for forming Egyptian fractions in which all denominators are odd. If a rational number ''x''/''y'' is a sum of odd unit fractions, :$\backslash frac\; =\; \backslash sum\backslash frac,$ then ''y'' must be odd. Conversely, it is known that whenever ''y'' is odd, every fraction ''x''/''y'' has a representation of this type in which all the unit fractions are different from each other. For instance, such a representation can be found by replacing the fraction ''x''/''y'' by ''Ax''/''Ay'' where ''A'' is a number of the form 35×3''i'' for a sufficiently large ''i'', and then expanding ''Ax'' as a sum of divisors of ''Ay'' (Breusch 1954; Stewart 1954). However, there is a simpler greedy algorithm that has successfully found Egyptian fractions in which all denominators are odd for all instances ''x''/''y'' (with odd ''y'') on which it has been tested: let ''u'' be the least odd number that is greater than or equal to ''y''/''x'', include the fraction 1/''u'' in the expansion, and continue in the same way with the remaining fraction ''x''/''y'' - 1/''u''. This method is called the odd greedy algorithm and the expansions it creates are called odd greedy expansions. Stein, Selfridge, Graham, and others have posed the question of whether the odd greedy algorithm terminates with a finite expansion for every ''x''/''y'' with ''y'' odd (Guy 1981). , this question remains open. Applying the odd greedy algorithm to a fraction with an even denominator produces an infinite series expansion. For instance Sylvester's sequence can be viewed as generated by the odd greedy expansion of 1/2. == Example == Let ''x''/''y'' = 4/23. 23/4 = 5 3/4; the next larger odd number is 7. So in the first step, we expand :4/23 = 1/7 + 5/161. 161/5 = 32 1/5; the next larger odd number is 33. So in the next step, we expand :4/23 = 1/7 + 1/33 + 4/5313. 5313/4 = 1328 1/4; the next larger odd number is 1329. So in the third step, we expand :4/23 = 1/7 + 1/33 + 1/1329 + 1/2353659. Since the final term in this expansion is a unit fraction, the process terminates with this expansion as its result. 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「Odd greedy expansion」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.