The abc Conjecture

For coprime positive integers a, b, c with a + b = c, and rad(n) the product of the distinct primes dividing n, the conjecture says: for every ε > 0, only finitely many triples have c > rad(abc)^{1+ε}.

Formalised by Oesterlé and Masser in 1985 from Frey's elliptic-curve observations; it implies asymptotic versions of Fermat, Mordell's conjecture (via Faltings-corollaries), and much of Diophantine approximation. Shinichi Mochizuki posted a 500-page proof in 2012 via his self-created Inter-universal Teichmüller theory; published in RIMS' journal in 2021. In 2018 Peter Scholze and Jakob Stix identified what they consider a severe, unfixable gap (Corollary 3.12); Mochizuki rejected their reading. As of 2026 the dispute persists: the LANA project's interim report on Lean formalisation could not reach consensus and remains stuck at precisely the Scholze–Stix point ('the wall'), while a majority of the community regards the argument as flawed. Status here: honestly open.