## On Catalan’s problem.(English)Zbl 0127.27102

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
Show Scanned Page ### MSC:

 11D61 Exponential Diophantine equations

### Keywords:

Catalan equation; exponential Diophantine equation

### Citations:

Zbl 0094.25702; Zbl 0116.02901
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