A note on the number of solutions of the Pillai type equation |ax - by|= k

Let a, b, k be fixed positive integers such that min{a, b} > 1and gcd(a, b) = 1, and let N '( a, b, k) denote the number of positive integer solutions (x, y) of the equation |a(x)- b(y)|= 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 epsilon with 0 < epsilon < 1, if k< min{a(1-epsilon), b(1-epsilon)} and max{a, b} > C(epsilon), where C(epsilon) is an effectively computable constant depending only on epsilon, then N '(a, b, k) <= 1. In particular, if k < min {a(1/15), b(1/15)}, then N '(a, b, k) <= 1except for N '(2, 3, 1) = N '(3, 2, 1) = 3. (c) 2021 Elsevier Inc. All rights reserved.