OFFSET
1,2
COMMENTS
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".
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.
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.
REFERENCES
A. H. Beiler, Recreations in the Theory of Numbers, Dover, NY, 1964, pp. 162-165.
L. E. Dickson, History of the theory of numbers, vol. II: Diophantine Analysis, reprint, Chelsea, New York, 1966, pp. 705-716.
R. K. Guy, Unsolved Problems in Number Theory, D1.
LINKS
Grahame Bennett, Nicuşor Minculete, and Marian Tetiva, Problem 11635: When Sums of Powers Determine Their Terms, Amer. Math. Monthly, 121 (2014), p. 89.
Tito Piezas III, Equal Sums of Like Powers and the Prouhet-Tarry-Escott (PTE) Problem
Tito Piezas III and Eric W. Weisstein, Multigrade Equation, MathWorld
Carlos Rivera, Puzzle 65.- Multigrade Relations, The Prime Puzzles & Problems Connection
Chen Shuwen, Equal Sums of Like Powers
EXAMPLE
1 + 5 + 8 + 12 = 26 = 2 + 3 + 10 + 11.
1^2 + 5^2 + 8^2 + 12^2 = 234 = 2^2 + 3^2 + 10^2 + 11^2.
1^3 + 5^3 + 8^3 + 12^3 = 2366 = 2^3 + 3^3 + 10^3 + 11^3.
1 + 12 = 5 + 8 = 2 + 11 = 3 + 10 = 13.
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Jonathan Sondow, Feb 07 2014
STATUS
approved