ABC conjecture for coprime integer triples

About 1 year old · traced to 2025

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 1Other wordings

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).

References

Primary source

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

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The abc conjecture for coprime integer triples

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