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Subject:	Number Theory
	» Computational Number Theory

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Posted	by:	maxal
	on:	August 5th, 2007

Problem Find six positive integers $x_1, x_2, \dots, x_6$ such that $x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$ or prove that such integers do not exist.

Euler's sum of powers conjecture states that for $k\geq 3$ the Diophantine equation $\sum_{i=1}^{n} a_i^k = b^k$ does not have solutions in positive integers as soon as $n<k.$ For $k=3$ it corresponds to a particular case of Fermat Last Theorem and hence is true. For $k=4$ and $k=5$ , counterexamples to the Euler's sum of powers conjecture were found by N. Elkies in 1986 and L. J. Lander, T. R. Parkin in 1966 respectively. For $k=6$ , no counterexamples are currently known.

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