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Existence or nonexistence of odd perfect numbers.

Update: Goes back at least to Nicomachus of Gerasa around 100 AD, according to http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html Nichomachus also asked about infinitude of perfect numbers.

(Goes back at least to Descartes 1638 http://mathworld.wolfram.com/OddPerfectNumber.html and arguably all the way back to Euclid.)
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Existence or nonexistence of odd perfect numbers. (Goes back at least to Descartes 1638 http://mathworld.wolfram.com/OddPerfectNumber.html and arguably all the way back to Euclid.)