Solution of Pillai's problem with Balancing numbers and powers of 2

Abstract

An open conjecture of Pillai asserts that the equation x^m m − yn = k in integersm, n, x, y, k with m ≥ 3, n ≥ 2, x ≥ 2, y ≥ 2 and k ̸= 0 has a finite numberof solutions. We use tools from Baker’s linear forms theory to solve the Pillaiproblem for Balancing numbers and powers of 2. More precisely, we find all theinteger numbers c which has at least two representations of the form Br − 2sfor non-negative integers r and s. The strategy of solution depends mainly onMatveev’s fundamental inequality and on an inequality due to A. Dujella and A.Peth¨o to reduce the very large bounds obtained by Matveev’s theorem .