On Lucas and Pell–Lucas numbers that are close to each other and their intersections

In this paper, we investigate the closeness between Lucas and Pell–Lucas numbers by extending the notion of closeness introduced by Chern and Cui to these two classical sequences. More precisely, we study the Diophantine inequalities [Formula: see text] and [Formula: see text], in non-negative integers [Formula: see text] and [Formula: see text] with [Formula: see text], and determine all their common solutions. To obtain our results, we apply Matveev’s theorem on linear forms in logarithms with properties of continued fractions and Legendre’s criterion.