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History and philosophy of mathematics, biographies of mathematicians, mathematics education, recreational mathematics, communication of mathematics.

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Examples of conjectures that were widely believed to be true but later proved false

Luzin's conjecture was widely believed to be false, until it was proven by Carleson in 1966. I'm citing from Lennart Carleson's biography: "In 1913 Luzin conjectured that if a function $f$ is square …
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