Intersections of Pell, Pell-Lucas Numbers and Sums of Two Jacobsthal Numbers

Abstract

We use tools from Baker’s linear forms theory to solve the two Diophantine equations Pk = Jn + Jm and Qk = Jn + Jm where {Pk}k≥0, {Qk}k≥0 and {Jk}k≥0 are the sequences of Pell, Pell-Lucas and Jacobsthal numbers, respectively. The strategy depends mainly on the properties of the three sequences and Matveev’s inequality which is an indispensable tool in the theory of linear forms. Also, we employ an inequality due to A. Dujella and A. Petho to reduce the too large bounds ¨ obtained by Matveev’s theorem.