abc 予想は正しいか？

任意の epsilon > 0 に対し、a + b = c を満たす互いに素な正整数の三つ組のうち、c が rad(abc) の 1+epsilon 乗を超えるものが有限個しかないかを決定せよ。

Decide whether, for every epsilon > 0, there are only finitely many coprime triples a + b = c of positive integers with c greater than the radical of abc raised to the power 1 + epsilon.

A claimed proof by Mochizuki was published in PRIMS in 2021, but Scholze and Stix identified what they argue is a gap in a central corollary, and the wider community has not accepted the proof. This question is therefore recorded as claimed and contested, not resolved.

解決したかどうかは、問いではなくアプローチごとに決まります。Atlas は判定しません。外部の判定者が何をしたかを記録します。

• Lean 4 formalization (formal-conjectures)

  形式証明

  到達状況

  未着手

  判定者

  the Lean 4 typechecker, against the pinned toolchain · 型検査

  定式化

  The abc conjecture, stated in Lean 4 with the radical defined in the same file.問いと同値Lean v4.33.1 + mathlib@0df444a3 (formal-conjectures@fc2696b2)検証対象を見る

  保たれているもの

  • The inequality and its quantifier structure

  加えられた仮定・条件

  • A Lean definition of the radical, checked in the same file against small examples

  Statement only.

• Inter-universal Teichmueller theory

  査読付きの証明

  到達状況

  主張あり（検証は未開始）

  判定者

  journal peer review plus the arithmetic geometry community; PRIMS accepted the papers, the wider community has not · 査読

  定式化

  Derive the inequality from a theory of arithmetic deformations developed for this purpose.問いと同値

  保たれているもの

  • Targets the conjecture itself, in the form used by the field

  弱まっているもの

  • The step under dispute (Corollary 3.12) is the one that produces the inequality; without it nothing is obtained

  加えられた仮定・条件

  • A large body of new theory that few researchers have been able to check

  Atlas records that an adjudication process exists and is contested; it does not rule on the mathematics.

  閉じた道

  • Scholze and Stix argue that the central corollary cannot be proved as stated, and no reconciliation has been accepted by both sides; the dispute is over a single identified step.無条件
    • otherhttps://www.math.uni-bonn.de/people/scholze/WhyABCisStillaConjecture.pdfP. Scholze, J. Stix, Why abc is still a conjecture (2018): the objection to Corollary 3.12

  主張

  • Shinichi Mochizuki（2021-01-20） · 検証が始まっている

• Consequences as indirect evidence

  専門家の合意

  到達状況

  判定者なし・判定がつかない

  判定者

  none: an implication is not a proof · 判定手続きなし

  定式化

  Note that the conjecture implies Fermat-type results and Mordell's conjecture, and treat the coherence of those consequences as support.問いより弱い

  保たれているもの

  • Shows the conjecture sits correctly among known theorems

  弱まっているもの

  • Implications point the wrong way: they cannot establish the conjecture

  加えられた仮定・条件

  • Nothing

  証拠

  • paperdoi:10.1155/S1073792891000144N. Elkies, ABC implies Mordell, Int. Math. Res. Not. 1991, 99-109

• repositoryhttps://github.com/google-deepmind/formal-conjectures@fc2696b2f863e642ef1452938b0f031f5af5bed3FormalConjectures/Wikipedia/ABC.lean:60
• otherhttps://www.claymath.org/library/The conjecture as posed by Oesterle (1988) and Masser (1985)

URI

https://atlasalt.com/q/6712d9c0-7e0b-4f91-809b-b2c616cf2893

登録

2026-09-17

最終レビュー

2026-09-19

次回レビュー期限

2027-09-19

版

3499fe7dc5a2

ライセンス

CC-BY-4.0

立場

record_only（Atlas は判定しない）