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I chose year 1900 because of:

"Hilbert's problems are twenty-three problems in mathematics published by German mathematician David Hilbert in 1900. The problems were all unsolved at the time, and several of them were very influential for 20th-century mathematics. Hilbert presented ten of the problems (1, 2, 6, 7, 8, 13, 16, 19, 21, and 22) at the Paris conference of the International Congress of Mathematicians, speaking on August 8 in the Sorbonne. The complete list of 23 problems was published later, most notably in English translation in 1902 by Mary Frances Winston Newson in the Bulletin of the American Mathematical Society."

I know about some of them, to mention:

1) Legendre´s conjecture

2) Polignac´s conjecture

3) Bunyakovsky conjecture

4) Riemann hypothesis

5) Are there any odd perfect numbers?

6) Goldbach's conjecture

7) Are there infinitely many Mersenne primes?

8) Brocard's problem

But, I am not so sure is there a huge number of mathematical unsolved problems posed prior to the year 1900 or just a "few" of them?

Can you list some of them in your answer?

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    $\begingroup$ The Twin Prime Conjecture and the Inverse Galois Problem are two examples. $\endgroup$ – Nick R Jul 1 at 17:13
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    $\begingroup$ You mean those that became "famous"? The congruent number problem is old. But there were plenty of forgotten problems that people did not care enough about to solve. $\endgroup$ – Conifold Jul 1 at 17:34
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    $\begingroup$ The Navier-Stokes equations date from the mid-19th century. $\endgroup$ – Mark S Jul 1 at 19:48
  • $\begingroup$ Reminds me of an old Physics Prof. from my undergrad days who said (snarkily), "Here we have an integral. Mathematicians hide behind the integral sign and call that a solution, but us Physicists have to find the value of the integral to solve the problem" $\endgroup$ – Carl Witthoft Jul 2 at 12:06
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    $\begingroup$ I just noticed that the irrationality of $\gamma$, the Euler–Mascheroni constant, is not mentioned under any links provided so far. Wrote Euler in 1768:"This number seems also the more noteworthy because even though I have spent much effort in investigating it... the question remains of great moment, of what character the number O is and among what species of quantities it can be classified". In 1776 he asked the same question about $e^\gamma$, see Ufuoma $\endgroup$ – Conifold Jul 3 at 8:36
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Turns out this is discussed in detail over at MathOverflow. I'm in no way educated enough in this field to comment further, other than to mention that the top answer there states it's "Existence or nonexistence of odd perfect numbers," dating from Nicomachus of Gerasa around 100 AD.

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  • $\begingroup$ This might be better as a comment than an answer, unless you edit in the relevant information. $\endgroup$ – HDE 226868 Jul 3 at 1:41
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    $\begingroup$ @HDE226868 I quoted the top-rated answer. Figured that sufficed :-( $\endgroup$ – Carl Witthoft Jul 3 at 12:08

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