ISSN:
1572-9303
Keywords:
Catalan's equation
;
exponential diophantine equations
;
computational number theory
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract This paper begins with a short historical survey on Catalan's equation, namely xp-yq=1, where p andq are prime numbers and x, y are non-zero rational integers. It is conjectured that the only solution is the trivial solution 32-23=1. We prove that there is no non-trivial solution with p orq smaller than 30000. The tools to reach such a result are presented. A crucial role is played by a recent estimate of linear forms in two logarithms obtained by Laurent, Mignotte and Nestrenko. The criteria used are also quite recent. We give information on the enormous amount of computation needed for the verification.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1009701725510
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