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Euler's conjecture is a
conjecture related to
Fermat's Last Theorem which was proposed by
Leonhard Euler in
1769. It states that for every
integer n greater than 2, the sum of
n1
nth powers of positive integers cannot itself be an
nth power.
The conjecture was disproved by L. J. Lander[?] and T. R. Parkin[?] in 1966 when they found the following counterexample for n = 5:
 27^{5} + 84^{5} + 110^{5} + 133^{5} = 144^{5}.
In 1988, Noam Elkies[?] found a method to construct counterexamples for the n = 4 case. His smallest counterexample was the following:
 2682440^{4} + 15365639^{4} + 18796760^{4} = 20615673^{4}.
Roger Frye[?] subsequently found the smallest possible n = 4 counterexample by a direct computer search using techniques suggested by Elkies:
 95800^{4} + 217519^{4} + 414560^{4} = 422481^{4}.
No counterexamples for n > 5 are currently known.
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