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In number theory, Skewes' number is any of several extremely large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number ''x'' for which : where π is the prime-counting function and li is the logarithmic integral function. These bounds have since been improved by others: there is a crossing near . It is not known whether it is the smallest. ==Skewes' numbers== John Edensor Littlewood, who was Skewes' research supervisor, had proved in that there is such a number (and so, a first such number); and indeed found that the sign of the difference π(''x'') − li(''x'') changes infinitely often. All numerical evidence then available seemed to suggest that π(''x'') was always less than li(''x''). Littlewood's proof did not, however, exhibit a concrete such number ''x''. proved that, assuming that the Riemann hypothesis is true, there exists a number ''x'' violating π(''x'') < li(''x'') below : In , without assuming the Riemann hypothesis, Skewes was able to prove that there must exist a value of ''x'' below : Skewes' task was to make Littlewood's existence proof effective: exhibiting some concrete upper bound for the first sign change. According to George Kreisel, this was at the time not considered obvious even in principle. Although both Skewes' numbers are very large compared to most numbers encountered in mathematical proofs, neither is anywhere near as large as Graham's number. 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「Skewes' number」の詳細全文を読む スポンサード リンク
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