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%I #24 Sep 26 2023 09:53:58
%S 1,5,8,12,2,3,10,11,1,8,10,17,2,5,13,16,1,10,12,23,3,5,16,22
%N Primitive, symmetric octuples of distinct numbers a,b,c,d,x,y,z,w with 0<a<b<c<d and a<x<y<z<w<d such that a^k + b^k + c^k + d^k = x^k + y^k + z^k + w^k, for k = 1,2,3.
%C If a,b,c,d,x,y,z,w satisfies the (in)equalities in the definition, then so does the translate a-t,b-t,c-t,d-t,x-t,y-t,z-t,w-t, for t<a. So we say a,b,c,d,x,y,z,w is "primitive" if a=1. If a+d = b+c = x+w = y+z, we say a,b,c,d,x,y,z,w is "symmetric".
%C Bennett, Minculete, and Tetiva show that there do not exist distinct numbers a,b,c,x,y,z with 0<a<b<c and a<=x<y<z<=c such that a^k + b^k + c^k = x^k + y^k + z^k, for k = 1,2.
%C In this 6-term multigrade problem, if the restriction a<=x<y<z<=c is weakened to 0<x<y<z, a solution is 1,6,8,2,4,9, because 1^k + 6^k + 8^k = 2^k + 4^k + 9^k, for k = 1,2.
%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 162-165.
%D L. E. Dickson, History of the theory of numbers, vol. II: Diophantine Analysis, reprint, Chelsea, New York, 1966, pp. 705-716.
%D R. K. Guy, Unsolved Problems in Number Theory, D1.
%H Grahame Bennett, Nicuşor Minculete, and Marian Tetiva, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.121.01.083">Problem 11635: When Sums of Powers Determine Their Terms</a>, Amer. Math. Monthly, 121 (2014), p. 89.
%H Tito Piezas III, <a href="https://sites.google.com/view/tpiezas/019-equal-sums-of-like-powers-and-the-prouhet-tarry-escott-pte-problem">Equal Sums of Like Powers and the Prouhet-Tarry-Escott (PTE) Problem</a>
%H Tito Piezas III and Eric W. Weisstein, <a href="http://mathworld.wolfram.com/MultigradeEquation.html">Multigrade Equation</a>, MathWorld
%H Carlos Rivera, <a href="http://www.primepuzzles.net/puzzles/puzz_065.htm">Puzzle 65.- Multigrade Relations</a>, The Prime Puzzles & Problems Connection
%H Chen Shuwen, <a href="http://euler.free.fr/eslp/eslp.htm">Equal Sums of Like Powers</a>
%e 1 + 5 + 8 + 12 = 26 = 2 + 3 + 10 + 11.
%e 1^2 + 5^2 + 8^2 + 12^2 = 234 = 2^2 + 3^2 + 10^2 + 11^2.
%e 1^3 + 5^3 + 8^3 + 12^3 = 2366 = 2^3 + 3^3 + 10^3 + 11^3.
%e 1 + 12 = 5 + 8 = 2 + 11 = 3 + 10 = 13.
%Y Cf. A237435, A239066, A239067, A239068.
%K nonn,more
%O 1,2
%A _Jonathan Sondow_, Feb 07 2014
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