Generalized Wall-Sun-Sun Primes and Monogenic Power-Compositional Trinomials

For positive integers a and b, we let Un be the Lucas sequence of the first kind defined by U0=0, U1=1 and Un=aUn−1+bUn−2 for n≥2, and let π(m):=π(a,b)(m) be the period length of Un modulo the integer m≥2, where gcd(b,m)=1. We define an a,b-Wall-Sun-Sun prime to be a prime p such that π(p2)=π(p). When (a,b)=(1,1), such a prime p is referred to simply as a Wall-Sun-Sun prime. We say that a monic polynomial f(x)∈ℤ[x] of degree N is monogenic if f(x) is irreducible over ℚ and {1,θ,θ2,…,θN−1} is a basis for the ring of integers of ℚ(θ), where f(θ)=0. Let f(x)=x2−ax−b, and let s be a positive integer. Then, with certain restrictions on a, b and s, we prove that the monogenicity of f(xsn)=x2sn−axsn−b is independent of the positive integer n and is determined solely by whether s has a prime divisor that is an a,b-Wall-Sun-Sun prime. This result improves and extends previous work of the author in the special case b=1.