The weighted sum of two S-units being a square
Let S denote the set of all those integers which are divisible only by the given primes \(p_ 1,...,p_ s\). Let a,b\(\in {\mathbb{Z}}\) such that \(p_ i\nmid ab\) for \(i=1,...,s\) and (a,b) is squarefree. The aim of this paper is to prove that the diophantine equation (1) \(ax+by=z^ 2\) has only finitely many effectively computable solutions in x,y\(\in S\), \(z\in {\mathbb{Z}}\) with (x,y) is squarefree. Actually, this result is a special case of a theorem of \textit{S. V. Kotov} [Acta Arith 31, 219-230 (1976; Zbl 0352.12002)]. But here there are given very precise bounds for linear forms in p-adic logarithm of algebraic numbers, which enables one to compute all the solutions of (1) for given a,b and \(p_ 1,...,p_ s.\) Combining this result with the reduction technique given in the author's book [Algorithms for diophantine equations, CWI Tract 65 (1989; Zbl 0687.10013)] it is proven that (1) has for \(a=1\), \(b=\pm 1\) and \(p_ 1,...,p_ s=2,3,5,7\) exactly 388 solutions.
linear forms in p-adic logarithm of algebraic numbers
scientific article; zbMATH DE number 4175019
scientific article