"A felicitous but unproved conjecture may be of much more consequence for mathematics than the proof of many a respectable theorem."

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Given a conjecture, the best thing is to prove it. The second best thing is to disprove it. The third best thing is to prove that it is not possible to disprove it, since it will tell you not to waste your time trying to disprove it. That's what Gödel did for the Continuum Hypothesis.

— Saharon Shelah