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

The aim of this work is to present a method using the cyclic sequences ₖ\,\θt,k\ and \ψt,k \ in the finite fields F_ρ, with ρ a prime, that yield divisors of Mersenne, Fermat and Lehmer numbers. The transformations τₜ and σₜ are introduced which lead to the proof of the cyclic nature of the sequences \θt,k\ and \ψt,k\. Results on the roots of the $H(x)$-polynomials in F_ρ form the central theme of the study.