The Sum of Powers Conjecture was proposed by Euler in 1769. It states that:

“for all integers n and k greater than 1, if the sum of nkth powers of non-zero integers is itself a kth power, thenn is greater than or equal to k.”

This can be more concisely written as:

If , where n > 1 and a_{1}, a_{2}, …, a_{n}, b are non-zero integers, then n ≥ k.

Fermat’s Last Theorem is simply a special case of this conjecture: if, when n=2, a_{1}^{k} + a_{2}^{k} = b^{k}, then 2 ≥ k. Hence, this conjecture can be seen to represent an attempt to generalise Fermat’s Last Theorem.

Although it is true for n = 3, it was disproved for k = 4 and k = 5. Mathematicians still do not know whether it is true for k ≥ 6.

In 1966, L.J. Lander and T. R. Parkin disproved this conjecture in the shortest article in any ‘serious’ mathematical journal (Source), which consisted of merely 2 lines.

This counterexample for k = 5 was found by a direct computer search on a CDC 6600 (a “flagship mainframe supercomputer”).