Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k.
In symbols, if where n > 1 and are positive integers, then .
If the conjecture were true, it would be a generalization of Fermat's last theorem, which could be seen as the special case n = 2: if , then .
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Counterexamples
k = 5
The conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when they found the following counterexample for k = 5:
k = 4
In 1986, Noam Elkies found a method to construct counterexamples for the k = 4 case.^{[1]} His smallest counterexample was the following:
A particular case of Elkies' solution can be reduced to the identity,^{[2]}^{[3]}
where
This is an elliptic curve with one solution as v_{1} = −31/467. From this initial rational point, one can then compute an infinite number of v_{i}. Substituting v_{1} into the identity and removing common factors gives the numerical example cited above.
In 1988, Roger Frye subsequently found the smallest possible k = 4 counterexample by a direct computer search using techniques^{[citation needed]} suggested by Elkies:
Moreover, this solution is the only one with values of the variables below 1,000,000.
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