The ABC conjecture

Establish the ABC conjecture: For every ε > 0 there exists a constant Kε such that for any integers a, b, c with a + b = c, one has c ≤ Kε rad(abc)^{1+ε}, where rad(n) denotes the product of the distinct prime divisors of n.

Background

The authors include the ABC conjecture as a central statement in Diophantine analysis. As they note, it would imply Fermat’s Last Theorem for all sufficiently large exponents n, providing a unifying explanation for the rarity of large solutions to a + b = c when a, b, c share many prime factors.

Despite claims of proofs, the conjecture remains broadly treated as open within the number theory community, and the authors present it in its standard quantitative form.

References

Conjecture 12.1 (The ABC Conjecture). For every E > 0, there exists Ke such that for any integers a, b, c satisfying a + b = c, we have c ≤ Kerad(abc)1+€.

— Understanding Fermat's Last Theorem's Proofs  (2508.10362 - Qiu et al., 14 Aug 2025) in Appendix 12.2 (The ABC Conjecture)

The well-known abc conjecture of Masser and Oesterlé asserts that, for any λ < 1, there are only finitely many abc triples of exponent λ.

— Bounds on the exceptional set in the $abc$ conjecture  (2410.12234 - Browning et al., 2024) in Section 1. Introduction

Conjecture [ABC conjecture] For every positive real number $\varepsilon > 0$, there exist only finitely many triples of coprime integers $(a, b, c)$ such that $a+b = c$ and

c > rad(abc){1 + \varepsilon}.

Here, $rad(n) = \prod_{p|n} p$ is the product of all prime factors of $n$.

— Formalizing Mason-Stothers Theorem and its Corollaries in Lean 4  (2408.15180 - Baek et al., 2024) in Section 1, Introduction (Conjecture [ABC conjecture])

To study \mathcal P\cap \mathcal V, the odd primes (up to sign) that occur as \tau-values, we employ the celebrated $abc$ Conjecture of Masser and Oesterl e Exp.~694.

— ABC implies that Ramanujan's tau function misses almost all primes  (2603.29970 - Angdinata et al., 31 Mar 2026) in Section 1 (Introduction and statement of results), Conjecture (abc Conjecture)

The following tantalizing assertion was conjectured by David Masser and Joseph Oesterle (see ) and is known as the $abc$-conjecture: For each $\varepsilon>0$, there exists an absolute constant $C(\varepsilon)>0$, such that for all primitive triples of $a,b,c$ integers (i.e. triples of integers with $\text{gcd}(a,b,c)=1$) satisfying $$a+b=c,$$ one has $$\max{a},{b},{c}}\leq C(\varepsilon)\cdot \prod_{p|a\cdot b\cdot c}p{1+\varepsilon}$$ where the product is over all the prime number $p$ dividing $a\cdot b\cdot c$.

— Construction of Arithmetic Teichmuller Spaces IV: Proof of the abc-conjecture  (2403.10430 - Joshi, 2024) in Section 2.1 (The $abc$-conjecture)

(The abc conjecture) Given a fixed real number \varepsilon > 0, there exists a constant c_{\varepsilon} such that if a + b = c and \gcd(a, b, c) = 1, then c\leq c_{\varepsilon}\rad(abc){1+\varepsilon}."

— Note on the Exceptional Set in the ABC Conjecture  (2608.16764 - Carella, 17 Aug 2026) in Conjecture 1.1, Section 1 (Introduction)