A note on the number of solutions of the Pillai type equation [formula omitted

Abstract

Let a,b,k be fixed positive integers such that min⁡{a,b}>1 and gcd⁡(a,b)=1, and let N′(a,b,k) denote the number of positive integer solutions (x,y) of the equation |ax−by|=k. In this paper, using a lower bound of linear forms in two logarithms combined with some properties of convergents of irrational numbers, we prove the following two results: (i) If min⁡{a,b}≥85988, then N′(a,b,k)≤2. (ii) For any real number ε with 0<ε<1, if k<min⁡{a1−ε,b1−ε} and max⁡{a,b}>C(ε), where C(ε) is an effectively computable constant depending only on ε, then N′(a,b,k)≤1. In particular, if k<min⁡{a1/15,b1/15}, then N′(a,b,k)≤1 except for N′(2,3,1)=N′(3,2,1)=3.