On Catalan’s problem. (English) Zbl 0127.27102

Inkeri, K.

Acta Arith. 9, 285-290 (1964).

The following two theorems concerning the Diophantlne equation (1) \(x^p-y^q = 1\) are proved:
A. Suppose that \(p\) and \(q\) are primes \(> 3\), and \(p\equiv 3\pmod 4\). If \(q\) does not divide the class number \(h(p)\) of the quadratic field \(\mathbb Q(\sqrt{-p})\) and the equation (1) has a solution \(x, y\) in non zero integers, then \(p^q\equiv p\pmod{q^2}\), \(x\equiv 0\pmod{q^2}\) and \(y\equiv 1\pmod{q^{2p-1}}\).
B. Let \(p\) and \(q\) be primes with \(p\equiv q\equiv 3\pmod 4\), \(p > q > 3\). If \(q\) does not divide the class number \(h(p)\) of the field \(\mathbb Q(\sqrt{-p})\) and (1) has a solution \(x, y\) in non zero integers, then \(p^q\equiv p\pmod{q^2}\), \(q^p\equiv q\pmod{p^2}\), \(x\equiv 0\pmod{q^2}\), \(y\equiv 0\pmod{p^2}\) and \(x\equiv 1\pmod{p^{2q-1}}\) with \(y\equiv 1\pmod{q^{2p-1}}\).
From (1) is easily deduced that there exist integers \(u, v\) such that \[ x-1=p^{q-1} u^q,\quad (x^p-1)/(x-1) = pv^q. \] In the proof use is made of the well known identity \[ 4(x^p-1)/(x-1) =Y^2(x)-(-1)^{(p-1)/2}Z^2(x) \] and of results due to J. W. S. Cassels [Proc. Camb. Philos. Soc. 56, 97–103 (1960); corrigendum 57, 187 (1961; Zbl 0094.25702)] and M. Gut [Acta Arith. 8, 113–122 (1963; Zbl 0116.02901)]. As an application it is shown that (1) is not soluble in non zero integers for a fairly large number of pairs \(p, q\).

Reviewer: W. Ljunggren

MSC:

11D61

Exponential Diophantine equations

Keywords:

Catalan equation; exponential Diophantine equation

Citations:

Zbl 0094.25702; Zbl 0116.02901