On Diophantine equations involving Lucas sequences

Abstract

In this paper, we study the Diophantine equation $u_n=R(m)P(m)^{Q(m)}$,where $R, P$ and $Q$ are some polynomials (under weak assumptions) and $u_n$ is a Lucas sequence, thus the sequence $(u_n)_{n\geq 0}$ with characteristic polynomial $f(x) = x^2-ax-b$, i.e., $(u_n)_{n\geq 0}$ is the integral sequence satisfying $u_0=0, u_1=1$, and $u_n = au_{n-1} +bu_{n-2}$, for all integers $n\geq 2$. We suppose that this sequence is non degenerated.In this paper, we describe how a method based on $p$-adic valuations can be settled to this kind of equation. We found a upper bound for solutions of special case of this Diophantine equation in the form $F_n=km^m(m+1)$, where $k,m,n$ are any given positive integer.