A criterion on Catalan's equation
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.
scientific article; zbMATH DE number 770123