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A perfect number is a positive integer that is equal to the sum of its proper positive divisors.

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Can $k$ be arbitrarily large in the following equations?

(Note: This question was cross-posted from MSE per Dris's request.) Let $N = q^k n^2$ be an odd perfect number in Eulerian form. (That is, $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, …
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