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?
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."