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.
Ahmet Emin (Fri,) studied this question.