On the solutions of the Diophantine equation [formula omitted

Abstract

Let (Fn)n≥0 be the Fibonacci sequence given by F0=0,F1=1 and Fn+2=Fn+1+Fn,foralln≥0. In this paper, we find all positive integer solutions (m,n,a,k) of the Diophantine equation Fn±a(10m−1)9=k! with 1≤a≤9. Our proof requires lower bounds for nonzero linear forms in two logarithms of algebraic numbers both in the complex and p-adic cases and some computer calculations.