A generalization of the equation (2(k)-1) (3(l)-1)=5(m)-1

Abstract

Recently, Luca and Szalay solved the equation (2(k) - 1) (3(l )- 1) = 5(m) - 1. Motivated by this equation, we show that the solutions of the equation (F-n(k) -1) (F-n+1(l)-1) = F-n+2 (m)- 1 are (n, k, l, m) = (3, 2, 2, 2) and (5, 2, 1, 2). Here F-n is the n(th) Fibonacci number. To prove this, the main tools are linear forms in logarithm of algebraic numbers and the Lenstra-Lenstra-Lovasz (LLL) lattice basis reduction algorithm.