ABC conjecture for coprime integer triples

For every $\epsilon>0$ , let $C_\epsilon$ denote a constant depending only on $\epsilon$ . Consider nonzero integers $a,b,c$ satisfying $\gcd(a,b,c)=1$ and $a+b=c$ , and define

\operatorname{rad}(abc)=\prod_{p\mid abc}p.

ABC conjecture. For every $\epsilon>0$ , there is a constant $C_\epsilon$ such that

\max\{|a|,|b|,|c|\}\leq C_\epsilon\operatorname{rad}(abc)^{1+\epsilon}.

The conjecture is recalled as a conditional tool for studying uniform bounds on Diophantine tuples and related quantities.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

The abc conjecture for coprime integer triples
Let $(a,b,c)$ be a triple of coprime positive integers satisfying $a+b=c$ . The abc conjecture. For every positive real number $\epsilon$ , there exists a positive real number $C(\epsilon)$ such that
$c<C(\epsilon)\operatorname{rad}(abc)^{1+\epsilon}.$
This conjecture is a central reformulation of Szpiro's conjecture and has numerous consequences in Diophantine number theory; the supplied text gives no resolution.
source: Robin Zhang, “The abcd conjecture, uniform boundedness, and dynamical systems”, arXiv:2206.09725 (2024).

Sources & referencesView supporting material

Primary source

Ernie Croot and Chi Hoi Yip, “Diophantine tuples and product sets in shifted powers”, arXiv:2504.04354 (2026).

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ABC conjecture for coprime integer triples

Equivalent formulations 1

The abc conjecture for coprime integer triples

Primary source

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Solutions 0