On Pillai’s Problem With Balancing Numbers and Powers of 2

Abstract

An open conjecture of Pillai asserts that the equation x m − y n = k has a finite number of solutions if m, n, x, y, k are integers with m ≥ 3, n, x, y ≥ 2 and k = 0 ̸ is fixed. Baker’s theory of linear forms is employed to solve a variant of the Pillai problem for Balancing numbers and powers of 2. More precisely, all the integer numbers c which can be expressed in the form Br − 2 s for non-negative integers r and s in at least two ways are determined. The strategy of solution depends mainly on Matveev’s fundamental inequality and on a reduction theorem of A. Dujella and A. Petho.