Let $P_m$ and $E_m$ be the $m$-th Padovan and Perrin numbers respectively. Let $r,s$ be non-zero integers with $r\ge 1$ and $s\in \{-1,1\}$, let ${\left\{U_n\right\}}_{n\ge 0}$ be the generalized Lucas sequence given by $U_{n+2}=r{U}_{n+1}+s{U}_n$, with $U_0=0$ and $U_1=1.$ In this paper, we give effective bounds for the solutions of the following Diophantine equations $$P_m=U_n{U}_k\text{and}{E}_m=U_n{U}_k,$$ where $m$, $n$ and $k$ are non-negative integers. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell and balancing sequences.

In this paper, we find all Pell and Pell-Lucas numbers written in the form $-2^a-3^b+5^c$, in nonnegative integers $a$, $b$, $c$, with $0\le max\{a,b\}\le c$.

Let ${\left(a_n\right)}_{n\ge 1}$ be the sequence given by $a_1=1$ and $a_n=n^{a_{n-1}}$ for $n\ge 2$. In this paper, we show that the only solution of the equation$$a_1+\cdots +a_n=m^l$$is in positive integers $l\>1,m$ and $n$ is $m=n=1$.

Let $\alpha$ and $\beta$ are conjugate complex algebraic integers which generate Lucas or Lehmer sequences. We present an algorithm to search for elements of such sequences which have no primitive divisors. We use this algorithm to prove that for all $\alpha$ and $\beta$ with h$(\beta /\alpha )\le 4$, the $n$-th element of these sequences has a primitive divisor for $n\>30$. In the course of proving this result, we give an improvement of a result of Stewart concerning more general sequences.

Ce texte montre qu’en combinant le théorème fort des six exponentielles de D.Roy et la conjugaison complexe, on peut obtenir un certain nombre de cas particuliers de la conjecture forte des quatre exponentielles.