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abc-conjecture
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It ties addition to multiplication and implies dozens of other results, including an asymptotic Fermat's Last Theorem. Shinichi Mochizuki announced a proof in 2012 through his inter-universal Teichmüller theory. Scholze and Stix identified a step they consider a fatal gap, and most number theorists do not regard the conjecture as settled. The status here is open, with the dispute recorded rather than resolved.
Posed 1985, Joseph Oesterlé and David Masser.
What would settle it
Lean statement in formal-conjectures