A sextic counterexample to Euler's sum of powers conjecture

Importance: Medium ✭✭
Author(s): Euler, Leonhard P.
Recomm. for undergrads: yes
Posted by: maxal
on: August 5th, 2007
Problem   Find six positive integers $ x_1, x_2, \dots, x_6 $ such that $$x_1^6 + x_2^6 + x_3^6 + x_4^6 + x_5^6 = x_6^6$$ or prove that such integers do not exist.

Euler's sum of powers conjecture states that for $ k\geq 3 $ the Diophantine equation $ \sum_{i=1}^{n} a_i^k = b^k $ does not have solutions in positive integers as soon as $ n<k. $ For $ k=3 $ it corresponds to a particular case of Fermat Last Theorem and hence is true. For $ k=4 $ and $ k=5 $, counterexamples to the Euler's sum of powers conjecture were found by N. Elkies in 1986 and L. J. Lander, T. R. Parkin in 1966 respectively. For $ k=6 $, no counterexamples are currently known.

* indicates original appearance(s) of problem.