Makowski's even perfect number power-sum conjecture

Let $n$ be an even perfect number represented as $n=x^m+y^m$, where $x$ and $y$ are positive integers and $m$ is an integer with $m\geq 2$. Makowski's conjecture. Then $m=3$ and $n=28$. This conjecture proposes that the only even perfect number expressible as a sum of two equal positive integral powers of exponent at least two is $28=3^3+1^3$. The preceding argument establishes the cubic case but does not resolve higher exponents; the general statement remains open.