A237434 Primitive, symmetric octuples of distinct numbers a,b,c,d,x,y,z,w with 0

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

Offset: 1

Author

Jonathan Sondow, Feb 07 2014

Keywords

• nonn
• more

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

Bennett, Minculete, and Tetiva show that there do not exist distinct numbers a,b,c,x,y,z with 0

In this 6-term multigrade problem, if the restriction a<=x

Examples

			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.
		

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

Crossrefs

Cf. A237435, A239066, A239067, A239068.

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