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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Class number bounds and Catalan's equation


Author: Ray Steiner
Journal: Math. Comp. 67 (1998), 1317-1322
MSC (1991): Primary 11D41; Secondary 11R29
MathSciNet review: 1468945
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Abstract: We improve a criterion of Inkeri and show that if there is a solution to Catalan's equation

\begin{equation}x^p-y^q=\pm 1,\end{equation}

with $p$ and $q$ prime numbers greater than 3 and both congruent to 3 $(\mathrm{mod}\,4)$, then $p$ and $q$ form a double Wieferich pair. Further, we refine a result of Schwarz to obtain similar criteria when only one of the exponents is congruent to 3 $(\mathrm{mod}\,4)$. Indeed, in light of the results proved here it is reasonable to suppose that if $q\equiv 3$ $(\mathrm{mod}\,4)$, then $p$ and $q$ form a double Wieferich pair.


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Additional Information

Ray Steiner
Affiliation: Department of Mathematics, Bowling Green State University, Bowling Green, Ohio 43403
Email: steiner@math.bgsu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-98-00966-1
PII: S 0025-5718(98)00966-1
Keywords: Catalan's equation, class number bounds, algebraic number fields
Received by editor(s): March 17, 1997
Article copyright: © Copyright 1998 American Mathematical Society