Math Conjectures

abcabc Conjecture

OPENLandmarkConjectureProposed 1985 · Standard version

Canonical statement

For every ε>0\varepsilon>0, there is a constant Kε>0K_\varepsilon>0 such that, whenever coprime positive integers a,b,ca,b,c satisfy a+b=ca+b=c,
c<Kε rad⁡(abc)1+ε,rad⁡(n)=∏p∣np. c<K_\varepsilon\,\operatorname{rad}(abc)^{1+\varepsilon},\qquad \operatorname{rad}(n)=\prod_{p\mid n}p.
View source LaTeX
For every \(\varepsilon>0\), there is a constant \(K_\varepsilon>0\) such that, whenever coprime positive integers \(a,b,c\) satisfy \(a+b=c\),
\[
  c<K_\varepsilon\,\operatorname{rad}(abc)^{1+\varepsilon},\qquad
  \operatorname{rad}(n)=\prod_{p\mid n}p.
\]
NT-DIO-036:[Masser1985ABC][Granville1998ABC][Mochizuki2021IUTIV][ScholzeStix2018ABC]
Number theoryAlgebraic geometry

Notes

The abcabc conjecture bounds the size of a coprime sum a+b=ca+b=c in terms of the distinct primes dividing abcabc. Masser's 1985 problem list is an early printed source [Masser1985ABC], and the conjecture has consequences throughout Diophantine analysis [Granville1998ABC]. Mochizuki's published IUT papers claim a proof [Mochizuki2021IUTIV], but the decisive compatibility step remains disputed [ScholzeStix2018ABC]. The catalog therefore records the canonical conjecture as open and links the claim audit rather than treating publication alone as resolution.

References (4)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.