Euler's sum of powers conjecture

In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers $n$ and $k$ greater than 1, if the sum of $n$ many $k$ th powers of positive integers is itself a $k$ th power, then $n$ is greater than or equal to $k$ :

$a_{1}^{k}+a_{2}^{k}+\dots +a_{n}^{k}=b^{k}\implies n\geq k$

The conjecture represents an attempt to generalize Fermat's Last Theorem, which is the special case $n = 2$ : if $a_{1}^{k}+a_{2}^{k}=b^{k},$ then $2 \geq k$ .

Although the conjecture holds for the case $k = 3$ (which follows from Fermat's Last Theorem for the third powers), it was disproved for $k = 4$ and $k = 5$ . It is unknown whether the conjecture fails or holds for any value $k \geq 6$ .