We examine the congruences and iterate the digit sums of integer sequences. We generate recursive number sequences from triple and quintuple product identities. And we use second order recursions to determine the primality of special number systems.

Y. Bilu, G. Hanrot et P.M. Voutier ont montré que pour toute paire de Lucas ou de Lehmer $(\alpha ,\beta )$ et pour tout $n\>30$, les entiers, dits nombres de Lucas (ou de Lehmer) $u_n(\alpha ,\beta )$ admettaient un diviseur primitif. L’objet de ce papier est de compléter la liste des nombres de Lucas et de Lehmer défectueux donnée par P.M. Voutier, afin d’en avoir une liste exhaustive.

Let $a_{d-1},\cdots ,a_0\in \mathbb{Z}$, where $d\in \mathbb{N}$ and $a_0\ne 0$, and let $X={\left(x_n\right)}_{n=1}^{\infty }$ be a sequence of integers given by the linear recurrence $x_{n+d}=a_{d-1}{x}_{n+d-1}+\cdots +a_0{x}_n$ for $n=1,2,3,\cdots$. We show that there are a prime number $p$ and $d$ integers $x_1,\cdots ,x_d$ such that no element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined by the above linear recurrence is divisible by $p$. Furthermore, for any nonnegative integer $s$ there is a prime number $p\ge 3$ and $d$ integers $x_1,\cdots ,x_d$ such that every element of the sequence $X={\left(x_n\right)}_{n=1}^{\infty }$ defined as above modulo $p$ belongs to the set $\{s+1,s+2,\cdots ,p-s-1\}$.

Let ${\left(G_n\right)}_{n\ge 1}$ be a binary linear recurrence sequence that is represented by the Lucas sequences of the first and second kind, which are $\left\{U_n\right\}$ and $\left\{V_n\right\}$, respectively. We show that the Diophantine equation $G_n=B·(g^{lm}-1)/(g^l-1)$ has only finitely many solutions in $n,m\in {\mathbb{Z}}^{+}$, where $g\ge 2$, $l$ is even and $1\le B\le g^l-1$. Furthermore, these solutions can be effectively determined by reducing such equation to biquadratic elliptic curves. Then, by a result of Baker (and its best improvement due to Hajdu and Herendi) related to the bounds of the integral points on...