Experimental Mathematics

Algorithms for finding good examples for the abc and Szpiro conjectures

Abderrahmane Nitaj

Abstract

The radical 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

$${\log c\over \log \rad(abc)} \quad\hbox$$ and $$\quad {\log abc \over \log \rad(abc)}$$

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 $\Q$ with high Szpiro ratio

$$\sigma={\log|\Delta|\over\log N}$$,

where $\Delta$ is the discriminant and N is the conductor.

Article information

Source
Experiment. Math., Volume 2, Issue 3 (1993), 223-230.

Dates
First available in Project Euclid: 3 September 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1062620832

Mathematical Reviews number (MathSciNet)
MR1273410

Zentralblatt MATH identifier
0818.11050

Subjects
Primary: 11J25: Diophantine inequalities [See also 11D75]
Secondary: 11G05: Elliptic curves over global fields [See also 14H52] 11Y50: Computer solution of Diophantine equations

Citation

Nitaj, Abderrahmane. Algorithms for finding good examples for the abc and Szpiro conjectures. Experiment. Math. 2 (1993), no. 3, 223--230. https://projecteuclid.org/euclid.em/1062620832


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