Experimental Mathematics

Catalan's equation has no new solution with either exponent less than 10651

Maurice Mignotte and Yves Roy

Abstract

We consider Catalan's equation $x^p-y^q=1$ (where all variables are integers and $p,q$ are greater than $1$), which has the obvious solution $9-8=1$. Are there others? Applying old and new theoretical results to a systematic computation, we were able to improve on recent work of Mignotte and show that Catalan's equation has only the obvious solutions when $\min\{p,q\}<10651$. Two crucial tools used are a new result of Laurent, Mignotte, and Nesterenko on linear forms of logarithms, and a criterion obtained by W. Schwarz in 1994.

Article information

Source
Experiment. Math., Volume 4, Issue 4 (1995), 259-268.

Dates
First available in Project Euclid: 14 March 2003

Permanent link to this document
https://projecteuclid.org/euclid.em/1047674387

Mathematical Reviews number (MathSciNet)
MR1387692

Zentralblatt MATH identifier
0857.11012

Subjects
Primary: 11D61: Exponential equations
Secondary: 11J86: Linear forms in logarithms; Baker's method

Citation

Mignotte, Maurice; Roy, Yves. Catalan's equation has no new solution with either exponent less than 10651. Experiment. Math. 4 (1995), no. 4, 259--268. https://projecteuclid.org/euclid.em/1047674387


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