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Two $S$-unit equations with many solutions

S. Konyagin·K. Soundararajan
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Abstract

We show that there exist arbitrarily large sets $S$ of $s$ prime numbers such that the equation $a+b=c$ has more than $\exp(s^{2-\sqrt{2}-ε})$ solutions in coprime integers $a$, $b$, $c$ all of whose prime factors lie in the set $S$. We also show that there exist sets $S$ for which the equation $a+1=c$ has more than $\exp(s^{\frac 1{16}})$ solutions with all prime factors of $a$ and $c$ lying in $S$.

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S. Konyagin, K. Soundararajan. 2006-04-20. Two $S$-unit equations with many solutions. https://arxiv.org/abs/math/0604453

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