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 $n$ $k$ th powers of non-zero integers is itself a $k$ th power, then $n$ 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 $a 1, a 2, \dots, a n, b$ are non-zero integers, then $n \geq k$ .

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 $2 \geq k. Hence, this conjecture can be seen to represent an attempt to generalise Fermat’s Last Theorem.$

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 $k \geq 6$ .

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.

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

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