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.