On the Diophantine Equation $${L^2_m}+{L^2_n}=2^a$$

Abstract

This study researches numbers that are powers of two and can be represented as the sum of the squares of any two Lucas numbers. We apply Baker's theory of linear forms in logarithms of algebraic numbers, combined with a variation of the Baker--Davenport reduction method, to solve the Diophantine equation $$L^2_m+L^2_n=2^a$$, where $$m$$, $$n$$ and $$a$$ are positive integers, as presented in this study.