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孪生素数猜想.
孪生素数猜想(英语:Twin prime conjecture)是数论中的一个未解决问题。这个猜想正式由希尔伯特在1900年国际数学家大会的报告上第8个问题中提出,可以这样描述:
其中,素数对(p, p + 2)称为孪生素数。
在1849年,阿尔方·德·波利尼亚克提出了一般的猜想:对所有自然数k,存在无穷多个素数对(p, p + 2k)。k = 1的情况就是孪生素数猜想。
哈代-李特尔伍德猜测
1921年,英国数学家哈代和李特尔伍德提出了以下的猜想:设
为前N个自然数里孪生素数的个数。那么
![{\displaystyle \pi _{2}(N)\approx \int _{2}^{N}{\frac {\mathrm {d} t}{(\ln t)^{2))}\approx 2C_{twin}\ {\frac {N}{\ln ^{2}N))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19041ad4536e0f9ffcb2f0cd9c48ea8a0f57c52d)
其中的常数
是所谓的孪生素数常数:
![{\displaystyle {\begin{aligned}C_{twin}&=\left(1-{\frac {1}{2^{2))}\right)\left(1-{\frac {1}{4^{2))}\right)\left(1-{\frac {1}{6^{2))}\right)\left(1-{\frac {1}{10^{2))}\right)\cdots \ =\prod _{p>2}\left(1-{\frac {1}{(p-1)^{2))}\right)\\&=0.6601618158468695739278121\ldots \end{aligned))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35e881224fe3b38d94e9321f196976bea2cb3f2d)
其中的p表示素数。
最新进展
2013年5月14日,《自然》杂志报道,数学家张益唐证明存在无穷多个素数对相差都小于7000万[1],可以用数式表示为
![{\displaystyle \liminf _{n\to \infty }(p_{n+1}-p_{n})<7\times 10^{7))](https://wikimedia.org/api/rest_v1/media/math/render/svg/b58e4f0ca6e0003c37032c498862b502d324b713)
此处“
”是第n个素数,“
”是素数间隙。
他的工作是对Goldston–Graham–Pintz–Yıldırım[2][3][4]的结果的重要改进。张益唐的论文已被《数学年刊》(Annals of Mathematics)于2013年5月21日接受[a][5][6][7]。陶哲轩随后开始了一个Polymath计划,由网上志愿者合作降低张益唐论文中的上限。[8]截至2014年4月,即张益唐提交证明之后一年,上限已降至246。[9]
参考资料
- 脚注
- ^ 2013年4月17日向《数学年刊》(Annals of Mathematics)投稿
- 引用
- ^ 连以婷. 他的「髮絲步」撞破數學界的「質數牆」 華人數學家張益唐破解百年數學謎題. TechNews 科技新报. 2013年6月28日 [2014-09-02]. (原始内容存档于2014-08-31) (中文).
- ^ D. Goldston, J. Pintz and C. Yildirim, Primes in tuples, I
- ^ D. Goldston, S. Graham, J. Pintz and C. Yildirim, Small gaps between primes and almost primes
- ^ D. Goldston, Y. Motohashi, J. Pintz and C. Yildirim, Small gaps between primes exist
- ^ 数学家张益唐破译“孪生素数猜想”. 新华网/腾讯新闻. 2013-05-18 [2013年5月19日]. (原始内容存档于2013-10-01) (中文(简体)).
- ^ First proof that infinitely many prime numbers come in pairs. Nature. 2013-05-14 [2013-06-02]. (原始内容存档于2015-08-14).
- ^ Zhang, Yitang. Bounded gaps between primes. Annals of Mathematics (Princeton University and the Institute for Advanced Study). 2014, 179 (3): 1121–1174 [2014-03-29]. (原始内容存档于2014-03-11) (英语). (需要订阅才能查看)
- ^ Tao, Terence. Polymath proposal: bounded gaps between primes. June 4, 2013 [2014-02-26]. (原始内容存档于2019-12-05).
- ^ Bounded gaps between primes. Polymath. [2014-03-27]. (原始内容存档于2013-06-20).
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