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Euler's conjecture

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 n-1 n-th powers of positive integers cannot itself be an n-th power.

The conjecture was disproved by L. J. Lander[?] and T. R. Parkin[?] in 1966 when they found the following counterexample for n = 5:

275 + 845 + 1105 + 1335 = 1445.

In 1988, Noam Elkies[?] found a method to construct counterexamples for the n = 4 case. His smallest counterexample was the following:

26824404 + 153656394 + 187967604 = 206156734.

Roger Frye[?] subsequently found the smallest possible n = 4 counterexample by a direct computer search using techniques suggested by Elkies:

958004 + 2175194 + 4145604 = 4224814.

No counterexamples for n > 5 are currently known.



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