2023/01/13 by Jones, Lenny
#11R04 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2301.05566
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)∈ \mathbb Z[x] of degree N is monogenic if f(x) is irreducible over \mathbb Q and \1,θ,θ2,…, θN-1\ is a basis for the ring of integers of \mathbb 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.