MathLabs

Open problem, Arithmetic and number theory, posed 1985

abc conjecture

Open

For coprime positive integers aa, bb, cc with a+b=ca+b=c, and for every ε>0\varepsilon > 0, only finitely many such triples satisfy c>rad⁡(abc)1+ε,c > \operatorname{rad}(abc)^{1+\varepsilon}, where rad⁡(n)\operatorname{rad}(n) denotes the product of the distinct primes dividing nn.

Research frontier as of 2026

As of 2026 the abc conjecture is treated as open by the mathematical mainstream. Mochizuki's claimed IUT proof was published in PRIMS in 2021, but Scholze and Stix's 2018 report identifying a gap around Corollary 3.12 has not been resolved to the wider community's satisfaction, and formal-verification efforts such as Project LANA (using the Lean proof assistant) have so far been unable to confirm the disputed step. Outside a small circle centered on Kyoto, essentially no number theorist regards the conjecture as proved.

Best known results

  • Unconditional effective bounds (Stewart–Yu, 2001) relate cc and rad⁡(abc)\operatorname{rad}(abc), but remain far weaker than the conjectured polynomial-type bound.
  • The polynomial analogue, the Mason–Stothers theorem, is proved unconditionally and models the kind of statement the conjecture makes for integers.
  • Mochizuki's Inter-universal Teichmüller theory claims a full proof (preprints from 2012, published in PRIMS 2021), but Scholze and Stix's 2018 report identifies an unresolved gap, and the claim is not accepted by most number theorists outside Kyoto.

Tools and where they stop

ToolAchievedWhere it stops
Elementary and analytic effective bounds (Stewart–Tijdeman, Stewart–Yu)Unconditional but much weaker bounds relating cc and rad⁡(abc)\operatorname{rad}(abc)Leaves an exponential gap from the conjectured polynomial-type bound
Inter-universal Teichmüller theory (Mochizuki)Claims a full proof, published in PRIMS (2021)A purported gap (Scholze–Stix, 2018) around Corollary 3.12 remains unresolved; the claim is not accepted by most of the field, and formalization attempts (Project LANA) have not confirmed the step

Open questions

  • Is the abc conjecture true?
  • Can Mochizuki's IUT argument be repaired, refuted, or formally verified — and if not, what would a widely accepted proof or counterexample look like?

References

  1. Joseph Oesterlé (1988). Nouvelles approches du "théorème" de Fermat
  2. Peter Scholze, Jakob Stix (2018). Why abc is still a conjecture
  3. Shinichi Mochizuki (2021). Inter-universal Teichmüller theory IV: Log-volume computations and set-theoretic foundations