abc conjecture: c < K(ε) · rad(abc)^(1+ε) for coprime a + b = c
Fourteen years after the IUT claim there is no accepted proof and no unconditional method beyond Stewart–Yu's exponential bound, so 85: below the Millennium problems (92–96) but above the rank-two BSD question (82). Application value 13 is lower than RH's 30 because no algorithm depends on it. Leverage 74 is second only to RH in this area: dozens of Diophantine results (effective Mordell, Szpiro, Roth with effective constants) are conditional on it.
The problem
If a plus b equals c, the numbers cannot all be built from many repeated small primes.
- 1Take the distinct primes of a, b and c
- 2Multiply them to get the radical
- 3c is conjectured not much bigger than it
- 4Mochizuki's proof claim remains disputed
For a positive integer , the radical is the product of its distinct prime factors; for example . The (Masser and Oesterlé, 1985) states: for every there is a constant such that for all coprime positive integers with , In words: addition and multiplication interact in a restricted way, so the size of is controlled by the distinct primes involved. The version asks for an explicit, computable .
Path to understanding
foundations → problemThe research entries this problem rests on, in the order to read them. Each links to its explainer.
- Start with
- Step 2
- Step 3
- Step 4
- Elliptic curvesTheory · host
- Then
- abc conjecture: c < K(ε) · rad(abc)^(1+ε) for coprime a + b = cThe problem
State of the art
Open in the accepted literature; a claimed proof is contested. The best unconditional bound is exponential in a power of the radical: Stewart and Yu (2001) proved log c < κ·rad(abc)^(1/3)·(log rad(abc))^3 for an effective constant κ, far from c ≪ rad^(1+ε). The polynomial analogue is a theorem (Mason–Stothers). The largest known quality log c / log rad(abc) is about 1.6299 (Reyssat: 2 + 3^10·109 = 23^5). Mochizuki's 2012 inter-universal Teichmüller (IUT) papers claim a proof via Szpiro's conjecture; they were published in 2021 in PRIMS, a journal of his own institute of which he was chief editor (the journal says a committee excluding him handled them). Scholze and Stix (2018) argue that the step from Theorem 3.11 to Corollary 3.12 of IUT III does not work as written; Mochizuki rejects their reading. Kirti Joshi (arXiv:2403.10430, 2024–2025) claims a proof via arithmetic Teichmüller spaces, which Mochizuki criticized in March 2024. On 17 July 2026 Project LANA (Fumiharu Kato with Commelin, Hoshi, Kedlaya, Topaz and others) released an interim report that isolates a compatibility problem in the 3.11→3.12 step, says the team has not reconstructed a proof of it, and aims eventually at a Lean formalization; Joshi disputed the report on 30 July 2026. No proof has consensus acceptance and there is no formal verification or refutation.
Realistic applications
None directly. A proof, especially an effective one, would make many Diophantine results effective and would sharpen bounds in the study of elliptic curves and integer-point counting, but no cryptographic or engineering application depends on it.
What counts as solved
A proof (with or without explicit K(ε)) accepted by the number theory community after independent verification, or a counterexample family. Acceptance of the IUT proof by independent experts with the Scholze–Stix objection resolved would also count.
How it gets verified
Lean proofA formal proof checked by Lean 4 / Mathlib.
Statement: ABC.abc in Wikipedia/ABC.lean (Formal Conjectures pinned at commit e754a78 (17 October 2026)): for every ε > 0 only finitely many coprime triples a + b = c satisfy c > rad(abc)^(1+ε), with the radical defined in the file from Mathlib's prime factorizations; the variants abc.variants.lt_constant_mul (c < K_ε·rad(abc)^(1+ε)) and abc.variants.quality are stated alongside. A proof is accepted when it compiles against the pinned corpus and passes Comparator. For the inter-universal Teichmüller claim, a Lean formalization would settle the Scholze–Stix dispute and is the only machine-checkable route.
Claims under verification
+ add a claimWhere the work is
The bottleneck is mathematical acceptance or a new proof; computation only searches for counterexamples.
Bottlenecks
ranked by votes- #1TheoryAcceptance or independent proof
IUT is not verified by independent experts, the Scholze–Stix gap remains disputed, and the LANA project has not reconstructed the disputed step. No other approach goes beyond Stewart–Yu’s exponent 1/3 in the radical.
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Connected problems
- This problem is related to Finiteness of the Tate–Shafarevich group Ш(E/ℚ) for an elliptic curve of analytic rank 2
Progress
- Jul 17, 2026Project LANA interim reportsource
Isolates a compatibility problem in the disputed 3.11→3.12 step; no verdict and no reconstruction of the step yet.
- Oct 1, 2018Scholze–Stix reportsource
Identify an alleged gap at corollary 3.12 of IUT III.
- Aug 30, 2012Mochizuki posts IUT paperssource
Claimed proof of abc via inter-universal Teichmüller theory.
- Jan 1, 1985Oesterlé–Masser formulate the conjecturesource
Motivated by Szpiro's conjecture on elliptic curves (date approximate to the year).