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Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.

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Examples of eventual counterexamples

The story of Legendre's constant seems to qualify. Based on numerical evidence available at the time and the hardness of computing more digits, it was eminently reasonable to conjecture he was not jus …
20 votes

Examples of algorithms requiring deep mathematics to prove correctness

Many property testing algorithms in the dense graph model rely on the Szemerédi regularity lemma [Sze78], which (essentially) guarantees that every large enough graph can be divided into parts of roug …