# Euler's Sum of Powers Conjecture

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## Famous False Conjecture

No $n$th power can be the sum of fewer than $n$ $n$th powers.

## Refutation

- $144^5 = 27^5 + 84^5 + 110^5 + 133^5$

## Source of Name

This entry was named for Leonhard Paul Euler.

## Historical Note

This was proven false by the counterexample found by Leon J. Lander and Thomas R. Parkin in $1966$, when they were using a computer to hunt for $5$th powers which were the sum of $5$ $5$th powers.

In one of the $4$ solutions they found, one of the contributing $5$th powers was $0^5$.

It was at that point it was realised that here was a counterexample to Euler's Sum of Powers Conjecture.

Since then, further counterexamples have been found, for various powers.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $144$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $144$