Algorithms for finding good examples for the $abc$ and Szpiro conjectures

Abstract

The radical $\mathrm{rad}\,n$ of an integer $n\ne0$ is the product of the primes dividing $n$. The $abc$-conjecture and the Szpiro conjecture imply that, for any positive relatively prime integers $a$, $b$, and $c$ such that $a+b=c$, the expressions $$ {\log c\over \log \mathrm{rad}(abc)} \quad\hbox{and}\quad {\log abc \over \log \mathrm{rad}(abc)} $$ are bounded. We give an algorithm for finding triples $(a,b,c)$ for which these ratios are high with respect to their conjectured asymptotic values. The algorithm is based on approximation methods for solving the equation $Ax^n-By^n=Cz$ in integers $x$, $y$, and $z$ with small $|z|$.

Additionally, we employ these triples to obtain semistable elliptic curves over $\mathbf{Q}$ with high Szpiro ratio $$ \sigma={\log|\Delta|\over\log N}, $$ where $\Delta$ is the discriminant and $N$ is the conductor.