Class number bounds and Catalan’s equation

Steiner, Ray

doi:10.1090/S0025-5718-98-00966-1

Class number bounds and Catalan’s equation
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by Ray Steiner PDF

Math. Comp. 67 (1998), 1317-1322 Request permission

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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