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- W4386763281 abstract "Let $f(x)in mathbb{Z}[x]$ be a monic polynomial of degree $N$ that is irreducible over $mathbb{Q}$. We say $f(x)$ is emph{monogenic} if $Theta={1,theta,theta^2,ldots ,theta^{N-1}}$ is a basis for the ring of integers $mathbb{Z}_K$ of $K=mathbb{Q}(theta)$, where $f(theta)=0$. If $Theta$ is not a basis for $mathbb{Z}_K$, we say that $f(x)$ is emph{non-monogenic}.Let $kge 1$ be an integer, and let $(U_n)$be the sequence defined by [U_0=U_1=0,qquad U_2=1 qquad text{and}qquad U_n=kU_{n-1}+(k+3)U_{n-2}+U_{n-3} qquad text{for $nge 3$}.] It is well known that $(U_n)$ is periodic modulo any integer $mge 2$, and we let $pi(m)$ denote the length of this period. We define a emph{$k$-Shanks prime} to be a prime $p$ such that $pi(p^2)=pi(p)$. Let $mathcal{S}_k(x)=x^{3}-kx^{2}-(k+3)x-1$ and $mathcal{D}=(k^2+3k+9)/gcd(3,k)^2$. Suppose that $knot equiv 3 pmod{9}$ and that $mathcal{D}$ is squarefree. In this article, we prove that $p$ is a $k$-Shanks prime if and only if $mathcal{S}_k(x^p)$ is non-monogenic, for any prime $p$ such that $mathcal{S}_k(x)$ is irreducible in $mathbb{F}_p[x]$. Furthermore, we show that $mathcal{S}_k(x^p)$ is monogenic for any prime divisor $p$ of $mathcal{D}$. These results extend previous work of the author on $k$-Wall-Sun-Sun primes." @default.
- W4386763281 created "2023-09-16" @default.
- W4386763281 creator A5003340477 @default.
- W4386763281 date "2023-09-15" @default.
- W4386763281 modified "2023-09-26" @default.
- W4386763281 title "On the monogenicity of power-compositional Shanks polynomials" @default.
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- W4386763281 doi "https://doi.org/10.7169/facm/2104" @default.
- W4386763281 hasPublicationYear "2023" @default.
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