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In number theory, the Pólya conjecture stated that "most" (i.e., 50% or more) of the natural numbers less than any given number have an ''odd'' number of prime factors. The conjecture was posited by the Hungarian mathematician George Pólya in 1919, and proved false in 1958 by C. Brian Haselgrove. The size of the smallest counterexample is often used to show how a conjecture can be true for many numbers, and still be false.〔.〕 ==Statement== Pólya's conjecture states that for any ''n'' (> 1), if we partition the natural numbers less than or equal to ''n'' (excluding 0) into those with an ''odd'' number of prime factors, and those with an ''even'' number of prime factors, then the former set has at least as many members as the latter set. (Repeated prime factors are counted the requisite number of times—thus 24 = 23 × 31 has 3 + 1 = 4 factors i.e. an even number of factors, while 30 = 2 × 3 × 5 has 3 factors, i.e. an odd number of factors.) Equivalently, it can be stated in terms of the summatory Liouville function, the conjecture being that : for all ''n'' > 1. Here, λ(''k'') = (−1)Ω(''k'') is positive if the number of prime factors of the integer ''k'' is even, and is negative if it is odd. The big Omega function counts the total number of prime factors of an integer. 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「Pólya conjecture」の詳細全文を読む スポンサード リンク
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