## Experimental Mathematics

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

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

#### 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