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Prouhet-Tarry-Escott Problem

Find two distinct sets of integers {a_1,...,a_n} and {b_1,...,b_n}, such that for k=1, ..., m,

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

The Prouhet-Tarry-Escott problem is therefore a special case of a multigrade equation. Solutions with n=m+1 are said to be "ideal" and are of interest because they are minimal solutions of the problem (Borwein and Ingalls 1994).

The smallest symmetric ideal solutions for m=9 was found by Borwein et al. (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)

as well as the second solution

(-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)

The previous smallest known symmetric ideal solution, found by Letac in the 1940s, is

(-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)

In 1999, S. Chen found the first ideal solution with m>=10,

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)

which is true for k=1, 2, ..., 11.

See also

Multigrade Equation

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References

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.

Referenced on Wolfram|Alpha

Prouhet-Tarry-Escott Problem

Cite this as:

Weisstein, Eric W. "Prouhet-Tarry-Escott Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Prouhet-Tarry-EscottProblem.html

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