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Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive terms in the resulting sequence, and so forth. The statement is named after mathematician Norman L. Gilbreath who, in 1958, presented it to the mathematical community after observing the pattern by chance while doing arithmetic on a napkin.〔.〕 In 1878, eighty years before Gilbreath's discovery, François Proth had, however, published the same observations along with an attempted proof, which was later shown to be false.〔 == Motivating arithmetic == Gilbreath observed a pattern while playing with the ordered sequence of prime numbers :2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... Computing the absolute value of the difference between term ''n''+1 and term ''n'' in this sequence yields the sequence :1, 2, 2, 4, 2, 4, 2, 4, 6, 2, ... If the same calculation is done for the terms in this new sequence, and the sequence that is the outcome of this process, and again ''ad infinitum'' for each sequence that is the output of such a calculation, the following five sequences in this list are given by :1, 0, 2, 2, 2, 2, 2, 2, 4, ... :1, 2, 0, 0, 0, 0, 0, 2, ... :1, 2, 0, 0, 0, 0, 2, ... :1, 2, 0, 0, 0, 2, ... :1, 2, 0, 0, 2, ... What Gilbreath—and François Proth before him—noticed is that the first term in each series of differences appears to be 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「Gilbreath's conjecture」の詳細全文を読む スポンサード リンク
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