English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ ismonogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and\[\{1,\theta,\theta^2,\ldots, \theta^{N-1}\}\] is a basis for the ring ofintegers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$.  Let $k$ be a positive integer, and let $U_n:=U_n(k,-1)$ be the Lucas sequence$\{U_n\}_{n\ge 0}$ of the first kind defined by \[U_0=0,\quad U_1=1\quad\mbox{and} \quad U_n=kU_{n-1}+U_{n-2} \quad \mbox{ for $n\ge 2$}.\] A$k$-Wall-Sun-Sun prime is a prime $p$ such that \[U_{\pi_k(p)}\equiv 0\pmod{p^2},\] where $\pi_k(p)$ is the length of the period of $\{U_n\}_{n\ge0}$ modulo $p$.  Let ${\mathcal D}=k^2+4$ if $k\equiv 1 \pmod{2}$, and ${\mathcalD}=(k/2)^2+1$ if $k\equiv 0 \pmod{2}$. Suppose that $k\not \equiv 0 \pmod{4}$and ${\mathcal D}$ is squarefree, and let $h$ denote the class number of${\mathbb Q}(\sqrt{{\mathcal D}})$. Let $s\ge 1$ be an integer such that, forevery 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  $x^{2s^n}-kx^{s^n}-1$ is monogenic for all integers $n\ge 1$ if and only ifno prime divisor of $s$ is a $k$-Wall-Sun-Sun prime.

A Connection Between the Monogenicity of Certain Power-Compositional Trinomials and $k$-Wall-Sun-Sun Primes