A note on the diophantine equation \(ax^ m - by^ n = k\)
Let \(a,b,k,x,y\) be positive integers with \(x,y>1\), \(\text{gcd}(ax,by)=1\). The author considers the number \(N\) of integer solutions \((m,n)\) with \(m,n>1\) of the exponential diophantine equation \[ ax^ m-by^ n=k. \] It was known to \textit{W. J. LeVeque} [Am. J. Math. 74, 325-331 (1952; Zbl 0047.041)] that \(N\leq 1\) for \(a=b=k=1\) (Catalan's equation). \textit{Z.-F. Cao} [Kexue Tongbao 35, 558-559 (1990)] proved that \(N\leq 4\) for \(k\leq 2\). In general, \textit{T. N. Shorey} [Indagationes Math. 48, 353-358 (1986; Zbl 0603.10019)] has shown that \(N\leq 9\) if \(a\) \(x^ m>953 k^ 6\). The author proves the following results: (a) If \(\min(a,b)=k=1\), then \(N\leq 1\). (b) If \(a=b=1\) and \(\min(x,y)\geq 10^ 5\), then \(N\leq 2\). (c) If \(\min(x,y)\geq e^ e\), then \(N\leq 3\). Thereby it follows: (a) elementarily from the classical results of Birkhoff-Vandiver on primitive divisors of \(X^ r-Y^ r\), (b) from Mignotte-Waldschmidt's theorem on linear forms in two logarithms, and (c) from van der Poorten-Loxton's considerations of multiplicative relations in number fields.
multiplicative relations in number fields
scientific article; zbMATH DE number 78089