Because mathematics has been extremely well-developed in XX.-th and continues to do so in XXI.-th century and because there is an enormous number of open problems and conjectures and hypotheses posed from the beginning of XX.-th century till this time of ours, I am thinking that it would be nice to know something about conjectures that survived all the efforts of attack from the time of before the XX.-th century till the present day.

This question is posed with the hope that, in addition to the problems that we all know of that were most probably posed before the beginning of the XX.-th century, there are some problems that are unsettled but known only to some specialists in some branches of mathematics, so it would be nice to collect them here.

We all know about the problem of existence of an infinite number of Mersenne primes, about Goldbach˙s conjecture, about the problem of existence of an odd perfect number, also, all four of Landau´s problems were most probably posed before the beginning of the XX.-th century, also there is a famous conjecture of de Polignac, and problem of Brocard, and so many others, including the Riemann hypothesis.

Let us also take the interval (antiquity, 1900) to be open on the right side because Hilbert posed his problems in 1900 so some of his problems are not includable in the problems that you mention in your answer(s).