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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).

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    $\begingroup$ Check out the Millennium Prize problems. $\endgroup$
    – David G. Stork
    Commented Sep 4, 2018 at 18:49
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    $\begingroup$ In general, you are going to have a lot of famous number theory problems. As a general heuristic, if it is a number theory conjecture about which one gets crank emails and letters , it is probably in this category (with the only really obvious exception being the Collatz conjecture which is from the 1930s). $\endgroup$
    – JoshuaZ
    Commented Sep 4, 2018 at 23:43
  • $\begingroup$ Finding an optimal strategy for chess. Magnus Carlsen has produced some results. $\endgroup$
    – Gerry Myerson
    Commented Sep 5, 2018 at 0:47
  • $\begingroup$ @DavidG.Stork only the Riemann Hypothesis and Navier Stokes existence and smoothness predate the 20th century. The other 5 (now 4) all from the early to mid 20th century (with maybe P vs NP being hinted at by Gauss). $\endgroup$
    – Mark S
    Commented Jan 11, 2019 at 21:27

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The Inverse Galois Problem; determine whether a given group $G$ occurs as a Galois group over a given field $K$. According to the Introduction of the book by Jensen, Ledet, and Yui, "The Inverse Galois Problem was perhaps known to Galois.... The first systematic study of the Inverse Galois Problem started with Hilbert in 1892."

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Gauss' conjecture that there are infinitely many real quadratic fields with class number 1.

This is one of his conjectures on class numbers. So maybe it is too famous to be included in this list

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