A criterion on Catalan's equation
type: Item
instance of: scholarly article
title: A criterion on Catalan's equation
zbMATH Open document ID: 0829.11016
published in: Journal of Number Theory
publication date: 1996-01-28T00:00:00Z
review text: Let \(p\) and \(q\) be odd primes and consider Catalan's equation \(x^p - y^q = 1\) in \(x,y,p,q \in \mathbb{N}\). Let \(p - 1 = dl\) where \(l\) is odd and \(d\) is a power of 2. Assume \(l > 1\). Put \(\zeta = e^{2 \pi i/p}\). Let \(g\) be a primitive root modulo \(p\) and \(m = g^k \text{mod} p\). Let \(K = \mathbb{Q} (\xi)\) with \(\xi = \zeta + \zeta^m + \cdots + \zeta^{m^{l - 1}}\). Denote by \(h_K\) the class number of \(K\). In the present paper the author proves that Catalan's equation has no solutions if both \(q \nmid h_K\) and \(p^{q - 1} \not \equiv 1 \pmod {q^2}\) hold. This implies both of \textit{K. Inkeri}'s criteria [Acta Arith. 9, 285-290 (1964; Zbl 0127.271) and J. Number Theory 34, 142-152 (1990; Zbl 0699.10029)]. Moreover, the field involved is much smaller than the field that appears in Inkeri's second criterion. Combining the above criterion with earlier results the author shows that for any nontrivial solution of Catalan's equation \(\min \{p,q\} \geq 97\) holds.
Mathematics Subject Classification ID: 11D61
zbMATH DE Number: 770123
zbMATH Keywords: Catalan equation
zbMATH Keywords: Inkeri's criteria
zbMATH Keywords: exponential diophantine equation
author: Maurice Mignotte
reviewed by: István Gaál
MaRDI profile type: Publication
full work available at URL: jnth.1995.1070
OpenAlex ID: W2023380908
DOI: 10.1006/JNTH.1995.1070
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description: scientific article; zbMATH DE number 770123