TITLE: Examples of conjectures that were widely believed to be true but later proved false
QUESTION [45 upvotes]: It seems to me that almost all conjectures (hypotheses) that were widely believed by  mathematicians to be true were proved true later, if they ever got proved. Are there any notable exceptions?

REPLY [2 votes]: The longest-standing one of the sort is the "conjecture" that the parallel postulate can be proved using Euclid's first four postulates. I know that it is a far-fetched understanding of "conjecture". But, after all, it is something that people tried to prove by doing it for more than two thousand years, believing that it is true.