Existence or nonexistence of odd perfect numbers.
Update: Goes back at least to Nicomachus of Gerasa around 100 AD, according to J J O'Connor and E F Robertson. Nichomachus also asked about infinitude of perfect numbers.
(Goes back at least to Descartes 1638 https://mathworld.wolfram.com/OddPerfectNumber.html and arguably all the way back to Euclid.)
