Two Diophantine Approaches to the Irreducibility of Certain Trinomials

In the previous paper [FLSU], we determined the Galois groups associated with some polynomials constructed through the use of circulant matrices. In the process of determining the Galois groups, the irreducibility of the trinomial x + x +m was established, where p represents an odd prime and m an integer ≥ 2. The approach there was based on a method of Lebesgue [Le]. In this paper, we discuss two other approaches we discovered in our investigations, both relying on the nice work of Bilu, Hanrot, and Voutier [BHV] (with an appendix by Mignotte) on Lucas and Lehmer numbers. In the next section, we consider the more general polynomials tp(x) = x +bx+c where b and c are nonzero integers. There are four cases where reducibility is easily established: