On computing integral points of a Mordell curve – the method of Wildanger revisited

Abstract

We develop a new efficient algorithm for solving Mordell’s equation. Our method is based on Wildanger’s geometry of number approach. Major new ingredients come from Kummer theory and class field theory. This allows to enlarge the range of computations considerably. For explicit calculations, we used Magma and KANT.

Keywords:

• Mordell curve
• integral points
• Wildanger's method

Acknowledgments

Our special thanks goes to Oliver Voigt of TU Berlin who coordinated our extensive calculations there. The computation time for one polynomial in the first and second of the last three examples was up to 75 hours. The computations were carried out on a PC with “openSUSE Leap 42.1” and an AMD FX(tm)-6100 Six-Core Processor where only one kernel was used during each calculation. We thank the anonymous referees for valuable suggestions which helped to improve the original version of our paper.

Correction Statement

This article has been republished with minor changes. These changes do not impact the academic content of the article.

Notes

1 A new approach for solving Diophantine equations by Rafael von Känel and Benjamin Matschke with the title “Solving S-unit, Mordell, Thue, Thue-Mahler and generalized Ramanujan-Nagell equations via Shimura-Taniyama conjecture” can be found in arXiv: 1605.06079.

2 The cubic fields play a similar role as the reduced cubic forms in [1].

Additional information

Funding

Research supported in part by K115479 from the Hungarian National Foundation for Scientific Research and by the EFOP-3.6.1-16-00022 project. The project is co-financed by the European Union and European Social Fund.