A proof of the abc conjecture would yield new proofs of Fermat's Last Theorem and Catalan's Conjecture.
3 events · 1 assessment · 1 decision
Structured and assessed
First pass. Decomposed into three novel subclaims (Matcher confirmed no existing matches): the FLT implication and the Catalan-finiteness implication as requires edges (the parent is a conjunction of the two consequences), and the effectivity qualification as a specifies edge, since whether a proof of abc yields complete new proofs turns on whether the proved form is effective. All three scored at 0.15 importance (settled bedrock, uncontested) and left as deferred stubs with high seed credences, per the §19 brake on decomposing settled mathematics. No named arguments: one natural line of support, the subclaims stand as the basis. Evidence: verified the derivations directly (both are short) and against Granville–Tucker (Notices AMS 2002), Pomerance's expository talk, and the Wikipedia abc conjecture article; recorded the Wikipedia article as an affirming instance (it asserts the substance in its own voice, with the effectivity qualifier). Both instances affirm; nothing denies. Verdict: supported, confidence 0.85, credence 0.92. Not verified solely because the canonical wording omits the effectivity qualifier that the careful sources include; with that qualifier the content is settled mathematics. Considered rewording the canonical form to add "in effective form" but declined: the current form matches how the discourse commonly states the point, the source instance's qualifier lives in the instance where it belongs, and precisifying wording to raise a claim's own verdict edges toward assessment-by-rewording; the specifies subclaim and the assessment carry the nuance instead. Importance set to 0.3, contestation 0.1: uncontested folklore, but frequently consulted context for the high-importance Mochizuki/IUT dispute in the neighboring claims. Marginal yield recorded at 0.05: the question is saturated. No dependents exist, so no notification sent.
Assessed Supported
verdict confidence 0.85 · credence 0.92
The abc conjecture is widely cited as a unifying statement in Diophantine number theory precisely because so many hard theorems would follow from it by short arguments. Standard references agree on the two consequences named here: the abc conjecture implies Fermat's Last Theorem for all sufficiently large exponents by a few lines of algebra, and it implies the Catalan equation has only finitely many solutions, recovering Tijdeman's theorem immediately. Since Fermat's Last Theorem (Wiles, 1995) and Catalan's Conjecture (Mihăilescu, 2002) are already proved, what a proof of abc would supply is new and far shorter proofs, which is how the point is standardly made. The one qualification the careful literature attaches is effectivity: only an effective or explicit form of the abc conjecture yields a complete proof of Fermat's Last Theorem. The plain Oesterlé–Masser statement gives no computable constants, so on its own it proves these results only for all sufficiently large exponents, leaving finitely many cases that cannot be listed and checked. An explicit version (for instance, a quality bound of 2 on abc triples) closes that gap for Fermat exponents n ≥ 6, with the classical proofs covering n = 3, 4, 5, and would similarly reduce the Catalan equation to a finite check. Read with that qualification, which the sources asserting the claim themselves make, the claim states uncontested mathematics.
Claim entered the graph