Algorithms for Finding Good Examples for the abc and Szpiro Conjectures

Abstract

The radical rad n of an integer n ≠ 0 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 e such that a + b = c, the expressions

are bounded. We give an algorithm for finding triples (a, b, c) for which these ratios are high with respectto their conjectured asymptotic values. The algorithm is based on approximation methods for solving the equation Ax ⁿ – By ⁿ = C _z in integers x, y, and z with srnall |z|.

Additionally, we employ these triples to obtain semistable elliptic curves over Q with high Szpiro ratio

where Δ is the discriminant and N is the conductor.