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ShowingImportancePrizesTopicabc conjecture
A conjecture in number theory, due to David Masser and Joseph Oesterlé, concerning the relationship between the sum and product of coprime integers. Claims about its truth, proof, or disproof fall under this tag.
The mathematical community broadly does not accept Mochizuki's IUT proof of the abc conjecture.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Inter-universal Teichmüller theoryNumber theoryIUT Corollary 3.12 gapimportance · minorImportance 0.40, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Under Scholze and Stix's simplifying identifications, the argument for Corollary 3.12 yields only a trivial inequality
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →IUT Corollary 3.12 gapInter-universal Teichmüller theoryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Corollary 3.12 is the critical step in Mochizuki's claimed IUT proof of the abc conjecture.
VerifiedThe claim traces to reliable primary sources through a clear chain of evidence.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →IUT Corollary 3.12 gapInter-universal Teichmüller theoryNumber theoryimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Mochizuki's inter-universal Teichmüller theory papers constitute a valid proof of the abc conjecture.
ContradictedAvailable evidence weighs against the claim.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Inter-universal Teichmüller theoryNumber theoryimportance · majorImportance 0.80, from 0 to 1 · major: real consequence within a domain, actively argued. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Peer-reviewed publication of the IUT papers in Publications of RIMS establishes the proof's acceptance by the mathematical community.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Inter-universal Teichmüller theoryNumber theoryimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Several mathematicians, including Fesenko, Hoshi, Yamashita and Mok, have publicly affirmed that Mochizuki's IUT proof of the abc conjecture is correct.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Inter-universal Teichmüller theoryIUT Corollary 3.12 gapimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Public endorsements of Mochizuki's IUT proof come almost entirely from his RIMS colleagues and longstanding advocates.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Inter-universal Teichmüller theoryMathematicsimportance · minorImportance 0.35, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Scholze's zbMATH review of the published IUT papers maintains that the proof of Corollary 3.12 is incomplete.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →IUT Corollary 3.12 gapInter-universal Teichmüller theoryimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Scholze and Stix's 2018 report concluded that Mochizuki's IUT proof of the abc conjecture contains a serious, unfixable gap.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Inter-universal Teichmüller theoryIUT Corollary 3.12 gapimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
A proof of the abc conjecture would yield new proofs of Fermat's Last Theorem and Catalan's Conjecture.
SupportedEvidence favors the claim, but the chain is incomplete or the sources are secondary.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Number theoryFermat's Last TheoremCatalan's conjectureimportance · minorImportance 0.30, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Only an effective or explicit form of the abc conjecture yields a complete proof of Fermat's Last Theorem
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Fermat's Last TheoremNumber theoryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The abc conjecture implies the Catalan equation x^p − y^q = 1 has only finitely many solutions
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Number theoryCatalan's conjectureimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
The abc conjecture implies Fermat's Last Theorem for all sufficiently large exponents
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →Fermat's Last TheoremNumber theoryimportance · settledImportance 0.15, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
If Mochizuki's Corollary 3.12 holds, the abc conjecture follows from his remaining arguments.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →IUT Corollary 3.12 gapInter-universal Teichmüller theoryNumber theoryimportance · settledImportance 0.20, from 0 to 1 · settled: uncontested, so low even when much depends on it. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
Taylor Dupuy argues that Scholze and Stix's simplified argument does not refute Corollary 3.12.
UnassessedNo current assessment. Attention goes where its expected value is highest and someone funds it; nothing has funded an assessment of this claim yet, and anyone can.constitution →empirical · derivedA factual claim that rests on inference from other evidence rather than direct observation.constitution →IUT Corollary 3.12 gapInter-universal Teichmüller theoryNumber theoryimportance · minorImportance 0.25, from 0 to 1 · minor: narrow or largely settled, cheap to get right. Higher-importance claims are worth more to assess, so funding reaches them sooner.constitution →
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