On Catalan’s conjecture. (English) Zbl 0699.10029

Inkeri, K.

J. Number Theory 34, No. 2, 142-152 (1990).

The paper contains some results on Catalan’s equation (*) \(x^ p-y^ q=1\) in nonzero integers x, y and odd primes p, q. Let \(h_ m\) be the class number of the cyclotomic field \(K_ m={\mathbb{Q}}(e^{2\pi i/m}).\)
Theorem: Let (x,y,p,q) be a solution of (*). If \(q\nmid h_ p\), then \(x\equiv 0 (mod q^ 2)\) and \(p^ q\equiv p (mod q^ 2)\) and if \(p\nmid h_ q\), then \(y\equiv 0 (mod p^ 2)\) and \(q^ p\equiv q (mod p^ 2)\). In his proof, the author uses Cassels’ result q \(| x\), p \(| y\) and some arithmetic in the fields \(K_ p\), \(K_ q\). The author gives several applications of the theorem stated above and previous results [e.g., the author, Acta Arith. 9, 285-290 (1964; Zbl 0127.271)]. For instance, he proves that (*) has no solutions with \(p,q<89\).

Reviewer: J.-H.Evertse

MSC:

11D61

Exponential Diophantine equations

Keywords:

Catalan’s equation

Citations:

Zbl 0127.271

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