2022/11/27 by Lenny Jones, Jones, Lenny
Mathematics · Physics and Astronomy · #11R04 #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Theories and Applications #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2211.14834
openalex publication_date 2022/11/27 · openalex created_date 2022/12/10 · openalex updated_date 2026/07/28
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 k be a positive integer, and let Un:=Un(k,-1) be the Lucas sequence \Un\n≥ 0 of the first kind defined by U0=0, U1=1 and Un=kUn-1+Un-2 for n≥ 2. A k-Wall-Sun-Sun prime is a prime p such that Uπk(p)≡ 0 \pmodp2, where πk(p) is the length of the period of \Un\n≥ 0 modulo p. Let \mathcal D=k2+4 if k≡ 1 \pmod2, and \mathcal D=(k/2)2+1 if k≡ 0 \pmod2. Suppose that k\not ≡ 0 \pmod4 and \mathcal D is squarefree, and let h denote the class number of \mathbb Q(√(\mathcal D)). Let s≥ 1 be an integer such that, for every odd prime divisor p of s, \mathcal D is not a square modulo p and gcd(p,h\mathcal D)=1. In this article, we prove that x2sn-kxsn-1 is monogenic for all integers n≥ 1 if and only if no prime divisor of s is a k-Wall-Sun-Sun prime.