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 n kth 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 a1, a2, …, an, b are non-zero integers, then n ≥ k.
Fermat’s Last Theorem is simply a special case of this conjecture: if, when n=2, a1k + a2k = bk, 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.
This counterexample for k = 5 was found by a direct computer search on a CDC 6600 (a “flagship mainframe supercomputer”).