A perfect number is the sum of its (positive) divisors (excluding itself). I am wondering if a square could be a perfect number.
If it is an odd square, then, excluding itself, it has an even number of divisors which are odd. Adding them together yields an even sum. Therefore, an odd square could not be a perfect number.
What about the even square case? Every help is appreciated.
Best Answer
No.
The Euclid-Euler theorem states that any even perfect number $n$ (we don't know whether there are any odd ones) is of the form $$ n = 2^{k-1}(2^k - 1) $$ with $2^k - 1$ prime, and furthermore that any $n$ of that form is perfect (this last part is relatively easy to prove, but it is the former part you need). This is clearly not a square, since $2^k - 1$ is strictly larger than $2^{k-1}$ and prime.