A perfect number is a positive integer that is equal to the sum of its proper divisors, and all perfect numbers are even.
A sum of two squares is an integer that is the sum of two squares integers.
2026-03-26 04:20:03.1774498803
Show that if n is an even perfect number then n is not the sum of two squares.
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By a theorem of Euler, all even perfect numbers are of the form $2^{n-1}(2^n-1)$, where $2^n-1$ is a (Mersenne) prime. But $2^n-1\equiv3\mod4$ for all perfect numbers ($n\ge2$), and it appears only once in the prime factorisation, so by the sum of two squares theorem, the perfect number is not expressible as the sum of two squares.