Euler’s Conjecture

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 \sum _{i=1}^{n}a_{i}^{k}=b^{k}, where n > 1 and a1, a2, …, an, b are non-zero integers, then nk.

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.

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.

Screen Shot 2016-09-06 at 12.45.09 PM.png

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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