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 *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”).

CDC 6600 | Source: Wikipedia

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