翻訳と辞書 Words near each other ・ Poliellaspis ・ Poliellina ・ Poliespo ・ Poliesportiu d'Andorra ・ Polietes ・ Polietina ・ Poliez-le-Grand ・ Poliez-Pittet ・ Poliforum Chiapas ・ Poliforum Zamna ・ Polifórum Benito Juárez ・ Poligeenan ・ Polignac ・ Polignac (card game) ・ Polignac Memorandum ・ Polignac's conjecture ・ Polignac, Charente-Maritime ・ Polignac, Haute-Loire ・ Polignano a Mare ・ Polignano a Mare railway station ・ Poligny ・ Poligny, Aube ・ Poligny, Hautes-Alpes ・ Poligny, Jura ・ Poligny, Seine-et-Marne ・ Poligné ・ Poligono di Furbara ・ Poligono Industrial ・ Poligus ・ Polihale State Park Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Polignac's conjecture ： ウィキペディア英語版 Polignac's conjecture In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states: :For any positive even number ''n'', there are infinitely many prime gaps of size ''n''. In other words: There are infinitely many cases of two consecutive prime numbers with difference ''n''.〔, p. 112〕 The conjecture has not yet been proven or disproven for a given value of ''n''. In 2013 an important breakthrough was made by Zhang Yitang who proved that there are infinitely many prime gaps of size ''n'' for some value of ''n'' < 70,000,000.〔 〕 Later that year, James Maynard announced a related breakthrough which proved that there are infinitely many prime gaps of some size less than or equal to 600. As of April 14, 2014, one year after Zhang's announcement, according to the Polymath project wiki, ''n'' has been reduced to 246.〔(【引用サイトリンク】title=Bounded gaps between primes )〕 Further, assuming the Elliott–Halberstam conjecture and its generalized form, the Polymath project wiki states that ''n'' has been reduced to 12 and 6, respectively.〔(【引用サイトリンク】title=Bounded gaps between primes )〕 For ''n'' = 2, it is the twin prime conjecture. For ''n'' = 4, it says there are infinitely many cousin primes (''p'', ''p'' + 4). For ''n'' = 6, it says there are infinitely many sexy primes (''p'', ''p'' + 6) with no prime between ''p'' and ''p'' + 6. Dickson's conjecture generalizes Polignac's conjecture to cover all prime constellations. == Conjectured density == Let $\backslash pi\_n(x)$ for even ''n'' be the number of prime gaps of size ''n'' below ''x''. The first Hardy–Littlewood conjecture says the asymptotic density is of form :$\backslash pi\_n(x)\; \backslash sim\; 2\; C\_n\; \backslash frac\; \backslash sim\; 2\; C\_n\; \backslash int\_2^x$ where ''C''''n'' is a function of ''n'', and $\backslash sim$ means that the quotient of two expressions tends to 1 as ''x'' approaches infinity.〔.〕 ''C''2 is the twin prime constant :$C\_2\; =\; \backslash prod\_\; \backslash frac\; \backslash approx\; 0.66016\; 18158\; 46869\; 57392\; 78121\; 10014\backslash dots$ where the product extends over all prime numbers ''p'' ≥ 3. ''Cn'' is ''C''2 multiplied by a number which depends on the odd prime factors ''q'' of ''n'': :$C\_n\; =\; C\_2\; \backslash prod\_\; \backslash frac.$ For example, ''C''4 = ''C''2 and ''C''6 = 2''C''2. Twin primes have the same conjectured density as cousin primes, and half that of sexy primes. Note that each odd prime factor ''q'' of ''n'' increases the conjectured density compared to twin primes by a factor of $\backslash tfrac$. A heuristic argument follows. It relies on some unproven assumptions so the conclusion remains a conjecture. The chance of a random odd prime ''q'' dividing either ''a'' or ''a'' + 2 in a random "potential" twin prime pair is $\backslash tfrac$, since ''q'' divides 1 of the ''q'' numbers from ''a'' to ''a'' + ''q'' − 1. Now assume ''q'' divides ''n'' and consider a potential prime pair (''a'', ''a'' + ''n''). ''q'' divides ''a'' + ''n'' if and only if ''q'' divides ''a'', and the chance of that is $\backslash tfrac$. The chance of (''a'', ''a'' + ''n'') being free from the factor ''q'', divided by the chance that (''a'', ''a'' + ''2'') is free from ''q'', then becomes $\backslash tfrac$ divided by $\backslash tfrac$. This equals $\backslash tfrac$ which transfers to the conjectured prime density. In the case of ''n'' = 6, the argument simplifies to: If ''a'' is a random number then 3 has chance 2/3 of dividing ''a'' or ''a'' + 2, but only chance 1/3 of dividing ''a'' and ''a'' + 6, so the latter pair is conjectured twice as likely to both be prime. 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「Polignac's conjecture」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.