On the diophantine equation \(ap^ x+bq^ y=c+dp^ zq^ w\)
It is shown that the title equation (an exponential diophantine equation in x, y, z, and w, where p, q, a, b, c, d are fixed positive integers with \((p,q)=1)\) can be essentially solved, by proving the existence of effectively computable upper bounds on the solutions. Use is made of theorems of \textit{J.-H. Evertse} [Compos. Math. 53, 225-244 (1984; Zbl 0547.10008), Corollary 1, p. 225], \textit{A. J. van der Poorten} [Transcend. Theory, Adv. Appl., Proc. Conf., Cambridge 1976, 29-57 (1977; Zbl 0367.10034), Theorem 1] and \textit{M. Waldschmidt} [Acta Arith. 37, 257-283 (1980; Zbl 0357.10017), Main result]. The author claims that all constants involved can be explicitly computed in terms of the parameters. Moreover, the large bounds may be reduced so as to allow the practical solution of the equation. Demonstrations are given by the special equations \((p,q,a,b,c,d)=(2,3,1,1,1,1)\) and (2,3,5,7,11,1).
existence of effectively computable upper bounds
scientific article; zbMATH DE number 4152464