ABC conjecture

ABC conjecture
George Bergman · GFDL 1.2 · source
Name	ABC conjecture
Field	Number theory
Proposed	1985
Proposer	Joseph Oesterlé and David Masser
Status	disputed

ABC conjecture is a conjecture in Number theory proposed in 1985 by Joseph Oesterlé and David Masser that relates the prime factors of three positive integers a, b, c satisfying a + b = c. It connects deep results in Diophantine geometry, Elliptic curves, and Arithmetic geometry and has influenced work on problems associated with Fermat's Last Theorem, Mordell's conjecture, and Langlands program directions.

Statement

The conjecture asserts that for every ε > 0 there exists a constant K(ε) such that for coprime positive integers a, b, c with a + b = c the inequality c < K(ε)·rad(abc)^{1+ε} holds, where rad(abc) denotes the product of distinct prime divisors of abc. This formulation connects to bounds in Baker's theorem-type transcendence estimates and to height inequalities on Arakelov theory-style arithmetic surfaces. Equivalent reformulations appear using exponential forms tied to Szpiro's conjecture and to discriminant–conductor relations in the context of Faltings's theorem and Shimura varieties.

Motivation and implications

Motivations came from attempting to unify observations about unusually large powers and small radical values in integer triples and from analogies with conjectures in Algebraic number theory and Modular forms. If true, the conjecture implies statements about the finiteness of solutions to certain families of Diophantine equations and yields effective refinements of results originally proved by techniques surrounding Gelfond–Schneider theorem and Thue–Siegel–Roth theorem. Consequences include streamlined proofs or conditional derivations of results related to Fermat's Last Theorem, the Mordell conjecture (proved by Gerd Faltings), and strong constraints on exceptional behavior in Bugeaud–Mignotte–Siksek-style exponential Diophantine analyses.

Examples and numerical evidence

Numerical investigations examine triples like (1, 8, 9), (2, 3, 5), and high-quality examples such as those built from Mersenne primes or sequences tied to Catalan's conjecture computations. Computational searches led by researchers connected to ABC@Home-style distributed projects catalogued high-quality triples, comparing c versus rad(abc) values and identifying record-holders used as test cases in papers by groups associated with CWI and university number theory groups. Evidence is mixed: many triples conform to the expected bound with modest K(ε), while exceptional triples exhibiting large "quality" raised interest from authors affiliated with Cambridge University and Princeton University who studied asymptotic distributions of radical sizes and linked data to heuristics inspired by Erdős and Pólya.

History and proofs attempts

The conjecture was formulated by Joseph Oesterlé and David Masser in the mid-1980s and quickly attracted attention from figures such as Jean-Pierre Serre, Gerd Faltings, and Enrico Bombieri. Over decades, partial advances built on work by Silverman, Szpiro, and researchers in Iwasawa theory produced conditional equivalences and implications. A major claim of proof by Shinichi Mochizuki using inter-universal Teichmüller theory sparked debate involving scholars at RIMS, Kyoto University, and reviewers including Go Yamashita-connected mathematicians; the claim led to extended verification efforts and discussions at institutions like Princeton University and ETH Zurich. The mathematical community has recorded extensive correspondence and seminars involving names such as Freeman Dyson, Don Zagier, and contributors from MSRI studying the claimed approaches.

Consequences and related conjectures

Assuming the conjecture yields immediate consequences for classical conjectures and theorems: it implies effective versions of results akin to Faltings's theorem and provides conditional proofs of statements tied to Mordell–Lang conjecture-type finiteness in special families. It is closely related to Szpiro's conjecture on conductor–discriminant relations for Elliptic curves, has connections to certain cases of the ABC@Home-motivated statistical heuristics, and influences expectations in Lang's conjectures about rational points on varieties. Further links appear with conjectures in Galois representations and speculative bridges to aspects of the Langlands correspondence in arithmetic contexts.

Partial results and conditional cases

Several unconditional partial results establish weakened bounds replacing the exponent 1+ε by larger constants or additional logarithmic factors; such work involves methods by M. N. Baker-style transcendence techniques and results of Iwaniec and Heath-Brown on exponential sums. Conditional proofs or equivalences derive from assuming statements like Szpiro's conjecture or certain cases of the ABC@Home heuristics; authors such as Joseph Silverman and Locherbach produced conditional implications when combining results from Arakelov theory and Galois module estimates. Computational verifications up to large search bounds by teams at University of Georgia, University of Bordeaux, and projects linked to CNRS support the conjecture numerically while highlighting challenging exceptional families that guide ongoing theoretical efforts.

Category:Number theory conjectures