Euler's Conjecture

It is closely related to Fermat's Last theorem.

Fermat's Last Theorem

http://en.wikipedia.org/wiki/Fermat's_Last_Theorem

Euler, in 1769 proposed that there are no sets of numbers such that

a^4 + b^4 + c^4 = d^4

or

a^5 + b^5 + c^5 + d^5 = e^5

The conjecture was disproved in 1966 by Lander and Parkin who found counterexample

for n=5:

27^5 + 84^5 + 110^5 + 133^5 = 144^5

Another counterexample was found by Noam Elkies in 1988:

2682440^4 + 15365639^4 + 18796760^4 = 20615673^4

Roger Frye who found the smallest possible n=4 solution

95800^4 + 217519^4 + 414560^4 = 422481^4

Try combinations for n > 5. No known solutions are found so far!