A237434 - OEIS

A237434

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.

1, 5, 8, 12, 2, 3, 10, 11, 1, 8, 10, 17, 2, 5, 13, 16, 1, 10, 12, 23, 3, 5, 16, 22

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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

Table of n, a(n) for n=1..24.

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

Cf. A237435, A239066, A239067, A239068.

Sequence in context: A314400 A314401 A355168 * A133522 A133269 A247245