Let $L(s, E)= {\mathop{\sum }\nolimits}_{n\geq 1} {a}_{n} {n}^{- s} $ be the $L$-series corresponding to an elliptic curve $E$ defined over $ \mathbb{Q} $ and $\mathbf{u} = \mathop{\{ {u}_{m} \} }\nolimits_{m\geq 0} $ be a nondegenerate binary recurrence sequence. We prove that if ${ \mathcal{M} }_{E} $ is the set of $n$ such that ${a}_{n} \not = 0$ and ${ \mathcal{N} }_{E} $ is the subset of $n\in { \mathcal{M} }_{E} $ such that $\vert {a}_{n} \vert = \vert {u}_{m} \vert $ holds with some integer $m\geq 0$, then ${ \mathcal{N} }_{E} $ is of density $0$ as a subset of ${ \mathcal{M} }_{E} $.

A Lehmer number is a composite positive integer n such that ϕ(n)|n − 1. In this paper, we show that given a positive integer g > 1 there are at most finitely many Lehmer numbers which are repunits in base g and they are all effectively computable. Our method is effective and we illustrate it by showing that there is no such Lehmer number when g ∈ [2, 1000].

Let n be a positive integer. In this paper, we consider the diophantine equation We prove that this equation has only the positive integer solutions (n, x, y, z) = (1, t, 1, 1), (1, t, 3, 2), (3, 2, 2, 2). Therefore we extend the work done by Leszczyński (Wiadom. Mat., vol. 3, 1959, pp. 37–39) and Makowski (Wiadom. Mat., vol. 9, 1967, pp. 221–224).