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I am looking for reasonably famous mathematical problems that were once open for more than twenty five years (preferably more than a hundred) and whose subsequent resolution (proof or counterexample) was contrary to prevailing expectation.

For example, if in the future, someone should say, produce an odd perfect number, show that there can only be finitely many Mersenne primes, or come up with a counterexample to the Riemann Hypothesis---that's what I mean by ``contrary to prevailing expectation.''

Were there any such celebrated problems?

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  • $\begingroup$ Nash embedding theorem comes to mind, though I don't know much about it's history, so it might not be 100 years... $\endgroup$
    – user810677
    Sep 20, 2020 at 3:31
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    $\begingroup$ Another one is the unsolvability of the general quintic. Poor Italians were trying that one for a while before Abel came along. $\endgroup$
    – user810677
    Sep 20, 2020 at 3:33
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    $\begingroup$ Oh and of course the king of them all: Gödel's incompleteness theorem. I'd say no other theorem has ever had such an immediate impact on math. All research efforts on finding a consistent and complete "theory of everything" vanished overnight. $\endgroup$
    – user810677
    Sep 20, 2020 at 3:36
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    $\begingroup$ The Pythagoran's believed irrational numbers didn't exist for a while. $\endgroup$
    – user810677
    Sep 20, 2020 at 3:40
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    $\begingroup$ @SenZen Yes; thank you---the Incompleteness Theorem certainly did put a damper on Hilbert's Program. $\endgroup$
    – DDS
    Sep 20, 2020 at 4:13

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How about Euclid's fifth postulate? According to Wikipedia:

"The postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually it was discovered that inverting the postulate gave valid, albeit different geometries."

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