Existence or nonexistence of odd perfect numbers.  

Update: Goes back at least to Nicomachus of Gerasa around 100 AD, according to <a href="https://web.archive.org/web/20100209232200/http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Perfect_numbers.html"> J J O'Connor and E F Robertson</a>.  Nichomachus also asked about infinitude of perfect numbers.

<strike>(Goes back at least to Descartes 1638 http://mathworld.wolfram.com/OddPerfectNumber.html and arguably all the way back to Euclid.)</strike>