斯奎斯数

在数论中，斯奎斯数（英語：Skewes' number）是指南非数学家斯坦利·斯奎斯（英语：Stanley Skewes）（Stanley Skewes）用以表示满足下式之最小自然数x的上界的極大數字。

\pi (x)>\operatorname {li} (x)

，其中 $\pi$ 表示素数计数函数， $li$ 则表示对数积分。经过数学家对这一上界的不断改进，目前发现在 $e^{727.95133}$ 附近有满足上式的自然数，不过仍不清楚这是否是最小的斯奎斯数。

大小

約翰·恩瑟·李特爾伍德於1914年證明確實存在斯奎斯數，而且還進一步證明了 $\pi (x)$ 和 ${li}(x)$ 兩個函數會交叉無數次，也就是有無窮個交叉點。然而不管代入什麼數字， $\pi (x)$ 都小於 ${li}(x)$ ，因此，可以知道x一定是比人們所能計算的數字都來得大的。

斯奎斯於1933年證明了其中一個上界（需要黎曼假設），又被稱作第一斯奎斯數：

e^{e^{e^{79}}}<10^{10^{10^{34}}}

（左為準確值，右為近似值）

斯奎斯又於1955年證明了另外一個上界（不需要黎曼假設），又被稱作第二斯奎斯數：

e^{e^{e^{e^{7.705}}}}<10^{10^{10^{964}}}

（左為準確值，右為近似值）

斯奎斯給出了具體的上界，以表明李特爾伍德說的斯奎斯數究竟有多大。雖然斯奎斯數比其他日常生活及數學證明中出現的大多數數字都來得大，但這個數仍然遠遠小於葛立恆數。

参见

素数定理

参考文献

Bays, C.; Hudson, R. H., A new bound for the smallest x with π(x) > li(x) (PDF), Mathematics of Computation, 2000, 69 (231): 1285–1296 [2012-09-09], MR 1752093, （原始内容存档 (PDF)于2008-09-11）
Brent, R. P., Irregularities in the distribution of primes and twin primes, Mathematics of Computation, 1975, 29 (129): 43–56, JSTOR 2005460, MR 0369287, doi:10.2307/2005460
Chao, Kuok Fai; Plymen, Roger, A new bound for the smallest x with $π(x) > li(x)$ , International Journal of Number Theory, 2005, 6 (03): 681–690, MR 2652902, arXiv:math/0509312 , doi:10.1142/S1793042110003125
Kotnik, T., The prime-counting function and its analytic approximations, Advances in Computational Mathematics, 2008, 29 (1): 55–70, doi:10.1007/s10444-007-9039-2
Lehman, R. Sherman, On the difference π(x) − li(x), Acta Arithmetica, 1966, 11: 397–410, MR 0202686
Littlewood, J. E., Sur la distribution des nombres premiers, Comptes Rendus, 1914, 158: 1869–1872
Skewes, S., On the difference π(x) − Li(x), Journal of the London Mathematical Society, 1933, 8: 277–283
Skewes, S., On the difference π(x) − Li(x) (II), Proceedings of the London Mathematical Society, 1955, 5: 48–70, MR 0067145
te Riele, H. J. J., On the sign of the difference π(x) − Li(x), Mathematics of Computation, 1987, 48 (177): 323–328, JSTOR 2007893, MR 0866118
Rosser, J. B.; Schoenfeld, L., Approximate formulas for some functions of prime numbers, Illinois Journal of Mathematics, 1962, 6: 64–94, MR 0137689
Saouter, Yannick; Demichel, Patrick, A sharp region where $π(x) - li(x)$ is positive, Mathematics of Computation, 2010, 79 (272): 2395–2405, MR 2684372, doi:10.1090/S0025-5718-10-02351-3
Zegowitz, Stefanie, On the positive region of $\pi (x)-\operatorname {li} (x)$ : 69 pp., 2010
Rubinstein, M.; Sarnak, P., Chebyshev's bias, Experimental Mathematics, 1994, 3 (3): 173–197, MR 1329368
Wintner, A., On the distribution function of the remainder term of the prime number theorem, American Journal of Mathematics (The Johns Hopkins University Press), 1941, 63 (2): 233–248, JSTOR 2371519, MR 0004255, doi:10.2307/2371519

大数

范例（按数字大小排列）

古戈爾 · 古戈尔普勒克斯 · 斯奎斯数 · 古戈爾普勒克斯普勒克斯（英语：Googolplexplex） · 莫澤數 · 葛立恆數 · TREE(3)（英语：TREE(3)） · SSCG(3)（英语：Friedman’s SSCG function） · 拉約數 · 超限数 · ∞

表达方法

记号	高德納箭號表示法 · 康威鏈式箭號表示法 · 斯坦豪斯-莫澤表示法 · 急成长阶层 · 缓成长阶层
运算	超运算（迭代冪次 · 五級運算） · 阿克曼函數

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