Sequences in finite fields yielding divisors of Mersenne, Fermat and Lehmer numbers, I

Abstract

The aim of this work is to present a method using the cyclic sequences $\{M_k\},\{\theta_{t,k}\}$ and $\{\psi_{t,k} \}$ in the finite fields $\mathbb{F}_\rho$, with $\rho$ a prime, that yield divisors of Mersenne, Fermat and Lehmer numbers. The transformations $\tau_t$ and $\sigma_t$ are introduced which lead to the proof of the cyclic nature of the sequences $\{\theta_{t,k}\}$ and $\{\psi_{t,k}\}$. Results on the roots of the $H(x)$-polynomials in $\mathbb{F}_\rho$ form the central theme of the study.