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Odd cubic perfect numbers divisible by 5

Posted: Sat Apr 01, 2017 7:31 pm UTC
by LucasBrown
Have they been shown to be nonexistent?

I know that the existence of odd perfect numbers is an open question, and I also know that so many constraints have been derived on them that they probably don't exist, but I'm presently concerned only with those that are also cubes and multiples of 5.

This arises in relation to this puzzle, for which this question is the only part I haven't solved yet.