2023/03/21 by Jones, Lenny · 1 citation
#11R04 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2303.11872
Let f(x)∈ \mathbb Z[x] be a monic polynomial of degree N that is irreducible over \mathbb Q. We say f(x) is monogenic if Θ=\1,θ,θ2,… ,θN-1\ is a basis for the ring of integers \mathbb ZK of K=\mathbb Q(θ), where f(θ)=0. If Θ is not a basis for \mathbb ZK, we say that f(x) is non-monogenic. Let k≥ 1 be an integer, and let (Un) be the sequence defined by U0=U1=0, U2=1 and Un=kUn-1+(k+3)Un-2+Un-3 for n≥ 3. It is well known that (Un) is periodic modulo any integer m≥ 2, and we let π(m) denote the length of this period. We define a k-Shanks prime to be a prime p such that π(p2)=π(p). Let \mathcal Sk(x)=x3-kx2-(k+3)x-1. Let \mathcal D=(k/3)2+k/3+1 if k≡ 0 \pmod3, and \mathcal D=k2+3k+9 otherwise. Suppose that k\not ≡ 3 \pmod9 and that \mathcal D is squarefree. In this article, we prove that p is a k-Shanks prime if and only if \mathcal Sk(xp) is non-monogenic, for any prime p such that \mathcal Sk(x) is irreducible in \mathbb Fp[x]. Furthermore, we show that \mathcal Sk(xp) is monogenic for any prime divisor p of k2+3k+9. These results extend previous work of the author on k-Wall-Sun-Sun primes.