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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 Mills' constant ： ウィキペディア英語版 Mills' constant In number theory, Mills' constant is defined as the smallest positive real number ''A'' such that the floor of the double exponential function :$\backslash lfloor\; A^\{3^\{n\}\}\; \backslash rfloor$ is a prime number, for all positive integers ''n''. This constant is named after William H. Mills who proved in 1947 the existence of ''A'' based on results of Guido Hoheisel and Albert Ingham on the prime gaps. Its value is unknown, but if the Riemann hypothesis is true, it is approximately 1.3063778838630806904686144926... . ==Mills primes== The primes generated by Mills' constant are known as Mills primes; if the Riemann hypothesis is true, the sequence begins :2, 11, 1361, 2521008887, 16022236204009818131831320183, 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499, ... . If ''ai'' denotes the ''i''th prime in this sequence, then ''ai'' can be calculated as the smallest prime number larger than $a\_^3$. In order to ensure that rounding $A^$, for ''n'' = 1, 2, 3, …, produces this sequence of primes, it must be the case that . The Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently large cubic numbers, which is sufficient to prove this inequality if we start from a sufficiently large first prime $a\_1$. The Riemann hypothesis implies that there exists a prime between any two consecutive cubes, allowing the ''sufficiently large'' condition to be removed, and allowing the sequence of Mills primes to begin at ''a''1 = 2. , the largest known Mills (re that rounding $A^$, for ''n'' = 1, 2, 3, …, produces this sequence of primes, it must be the case that . The Hoheisel–Ingham results guarantee that there exists a prime between any two sufficiently large cubic numbers, which is sufficient to prove this inequality if we start from a sufficiently large first prime $a\_1$. The Riemann hypothesis implies that there exists a prime between any two consecutive cubes, allowing the ''sufficiently large'' condition to be removed, and allowing the sequence of Mills primes to begin at ''a''1 = 2. , the largest known Mills (probable) prime (under the Riemann hypothesis) is :$\backslash displaystyle\; ((((((((((((2^3+3)^3+30)^3+6)^3+80)^3+12)^3+450)^3+894)^3+3636)^3+70756)^3+97220)^3+66768)^3+300840)^3+1623568,$ which is 555,154 digits long. 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「Mills' constant」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.