2023/02/20 by Lenny Jones, Jones, Lenny
Physics and Astronomy · Mathematics · #Advanced Mathematical Theories and Applications #Advanced Mathematical Identities #Analytic Number Theory Research
paper · pdf · doi:10.48550/arxiv.2302.10357
Let k≥ 1 be an integer, and let (Un) be the Lucas sequence of the first kind defined by U0=0, U1=1 and Un=kUn-1+Un-2 for n≥ 2. It is well known that (Un) is periodic modulo any integer m≥ 2, and we let π(m) denote the length of this period. A prime p is called a k-Wall-Sun-Sun prime if π(p2)=π(p). 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 \mathbb Q(θ), where f(θ)=0. If Θ is not a basis for \mathbb ZK, we say that f(x) is non-monogenic. Define \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 that \mathcal D is squarefree. In this article, we prove that p is a k-Wall-Sun-Sun prime if and only if \mathcal Fp(x)=x2p-kxp-1 is non-monogenic. This result, combined with previous work, shows that \mathcal Fp(x) is monogenic if p is a prime divisor of k2+4.