Solutions of the Diophantine Equations B r = J s + J t and C r = J s + J

Abstract

Let B r r ≥ 0 , J r r ≥ 0 , and C r r ≥ 0 be the balancing, Jacobsthal, and Lucas balancing numbers, respectively. In this paper, the diophantine equations B r = J s + J t and C r = J s + J t are completely solved. The solutions rely basically on Matveev’s theorem on linear forms in logarithms of algebraic numbers and a procedure of reducing the upper bound due to Dujella and Pethö.