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Prouhet-Tarry-Escott 问题

找到两个不同的整数集合 {a_1,...,a_n} 和 {b_1,...,b_n}, 使得对于 k=1, ..., m,

sum_(i=1)^na_i^k=sum_(i=1)^nb_i^k.
(1)

因此，Prouhet-Tarry-Escott 问题是多重次数方程的一个特例。解 n=m+1 被称为“理想”解，并且由于它们是问题的最小解而备受关注 (Borwein and Ingalls 1994)。

Borwein等人发现了最小的对称理想解 m=9 (Lisonek 2000),

(-313)^k+(-301)^k+(-188)^k+(-100)^k+(-99)^k+99^k+100^k+188^k+301^k+313^k =(-308)^k+(-307)^k+(-180)^k+(-131)^k+(-71)^k+71^k+131^k+180^k+307^k+308^k,
(2)

以及第二个解

(-515)^k+(-452)^k+(-366)^k+(-189)^k+(-103)^k+103^k+189^k+366^k+452^k+515^k =(-508)^k+(-471)^k+(-331)^k+(-245)^k+(-18)^k+18^k+245^k+331^k+471^k+508^k.
(3)

Letac 在 1940 年代发现的先前已知的最小对称理想解是

(-23750)^k+(-20667)^k+(-20449)^k+(-11857)^k+(-436)^k+436^k+11857^k+20449^k+20667^k+23750^k =(-23738)^k+(-20885)^k+(-20231)^k+(-11881)^k+(-12)^k+12^k+11881^k+20231^k+20885^k+23738^k.
(4)

1999 年，S. Chen 发现了第一个 m>=10 的理想解，

0^k+11^k+24^k+65^k+90^k+129^k+173^k+212^k+237^k+278^k+291^k+302^k =3^k+5^k+30^k+57^k+104^k+116^k+186^k+198^k+245^k+272^k+297^k+299^k,
(5)

这对于 k=1, 2, ..., 11 成立。

另请参阅

多重次数方程

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参考文献

Borwein, P. and Ingalls, C. "The Prouhet-Tarry-Escott Problem Revisited." Enseign. Math. 40, 3-27, 1994. http://www.cecm.sfu.ca/~pborwein/PAPERS/P98.ps.Chen, S. "The Prouhet-Tarry-Escott Problem." http://member.netease.com/~chin/eslp/TarryPrb.htm.Chernick, J. "Ideal Solutions of the Tarry-Escott Problem." Amer. Math. Monthly 44, 62600633, 1937.Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, pp. 709-710, 2005.Dorwart, H. L. and Brown, O. E. "The Tarry-Escott Problem." Amer. Math. Monthly 44, 613-626, 1937.Hahn, L. "The Tarry-Escott Problem." Problem 10284. Amer. Math. Monthly 102, 843-844, 1995.Hardy, G. H. and Wright, E. M. "The Four-Square Theorem" and "The Problem of Prouhet and Tarry: The Number P(k,j)." §20.5 and 21.9 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 302-306 and 328-329, 1979.Lisonek, P. "New Size 10 Solutions of the Prouhet-Tarry-Escott Problem." 21 Jun 2000. http://listserv.nodak.edu/scripts/wa.exe?A2=ind0006&L=nmbrthry&P=558.Shuwen, C. "Equal Sums of Like Powers." http://euler.free.fr/eslp/h12468.htm.Sinha, T. "On the Tarry-Escott Problem." Amer. Math. Monthly 73, 280-285, 1966.Sinha, T. "Some System of Diophantine Equations of the Tarry-Escott Type." J. Indian Math. Soc. 30, 15-25, 1966.Wright, E. M. "On Tarry's Problem (I)." Quart. J. Math. Oxford Ser. 6, 216-267, 1935.Wright, E. M. "The Tarry-Escott and the 'Easier' Waring Problem." J. reine angew. Math. 311/312, 170-173, 1972.Wright, E. M. "Prouhet's 1851 Solution of the Tarry-Escott Problem of 1910." Amer. Math. Monthly 102, 199-210, 1959.

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Prouhet-Tarry-Escott 问题

请引用为

Weisstein, Eric W. "Prouhet-Tarry-Escott 问题。" 来自 MathWorld——Wolfram 网络资源。 https://mathworld.net.cn/Prouhet-Tarry-EscottProblem.html

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