L. Jones

activity

20042023

most citedA new integral basis for the centre of the Hecke algebra of type A

3 citations · 11 across the 15 of their papers we have counts for

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12 papers · 1 filter

math.NT2023

On the monogenicity of power-compositional Shanks polynomials

Lenny Jones

Let $f (x) \in Z [x]$ be a monic polynomial of degree $N$ that is irreducible over $Q$ . We say $f (x)$ is \emph{monogenic} if $Θ = {1, θ, θ^{2}, \dots, θ^{N - 1}}$ is a b…

math.NT2023

Generalized Wall-Sun-Sun primes and monogenic power compositional trinomials

Lenny Jones

For positive integers $a$ and $b$ , we let $[U_{n}]$ be the Lucas sequence of the first kind defined by \[U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=aU_{n-1}+bU_{n-2} \quad \mbox{ fo…

math.NT2023

A new condition for $k$ -Wall-Sun-Sun primes

Lenny Jones

Let $k \geq 1$ be an integer, and let $(U_{n})$ be the Lucas sequence of the first kind defined by \begin{equation*}\label{Eq:Lucas} U_0=0,\quad U_1=1\quad \mbox{and} \quad U_n=kU_{n-1…

math.NT2022★ 2 cited

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

Lenny Jones

We say that a monic polynomial $f (x) \in Z [x]$ of degree $N$ is monogenic if $f (x)$ is irreducible over $Q$ and \[\{1,θ,θ^2,\ldots, θ^{N-1}\}\] is a basis for th…

math.NT2022★ 2 cited

The Irreducibility and Monogenicity of Power-Compositional Trinomials

Joshua Harrington, Lenny Jones

A polynomial $f (x) \in Z [x]$ of degree $N$ is called \emph{monogenic} if $f (x)$ is irreducible over $Q$ and ${1, θ, θ^{2}, \dots, θ^{N - 1}}$ is a basis for the ring…

math.NT2020

Characteristic Polynomials of Simple Ordinary Abelian Varieties over Finite Fields

Lenny Jones

We provide an easy method for the construction of characteristic polynomials of simple ordinary abelian varieties $A$ of dimension $g$ over a finite field ${\mathbb F}_q…

math.NT2019

Monogenic cyclotomic compositions

Joshua Harrington, Lenny Jones

Let $m$ and $n$ be positive integers, and let $p$ be a prime. Let $T (x) = Φ_{p^{m}} (Φ_{2^{n}} (x))$ , where $Φ_{k} (x)$ is the cyclotomic polynomial of index $k$ . In this article,…

math.NT2019

Monogenic trinomials with non-squarefree discriminant

Lenny Jones, Daniel White

For each integer $n \geq 2$ , we identify new infinite families of monogenic trinomials $f (x) = x^{n} + A x^{m} + B$ with non-squarefree discriminant, many of which have small Galois group. Thes…

math.NT2014★ 1 cited

On Primitive Covering Numbers

Lenny Jones, Daniel White

In 2007, Zhi-Wei Sun defined a \emph{covering number} to be a positive integer $L$ such that there exists a covering system of the integers where the moduli are distinct divisors o…

math.NT2014

Representing Integers as the Sum of Two Squares in the Ring $Z_{n}$

Joshua Harrington, Lenny Jones, Alicia Lamarche

A classical theorem in number theory due to Euler states that a positive integer $z$ can be written as the sum of two squares if and only if all prime factors $q$ of $z$ , with $q\e…

math.NT2011

Using Lucas Sequences to Generalize a Theorem of Sierpiński

Lenny Jones

In 1960, Sierpiński proved that there exist infinitely many odd positive integers $k$ such that $k \cdot 2^{n} + 1$ is composite for all positive integers $n$ . In this paper, we prove s…

math.NT2011

Let $ε \in {- 1, 1}$ . A sequence of prime numbers $p_{1}, p_{2}, p_{3}, ...$ , such that $p_{i} = 2 p_{i - 1} + ε$ for all $i$ , is called a {\it Cunningham chain} of the first or second kind, depen…

On the monogenicity of power-compositional Shanks polynomials

Generalized Wall-Sun-Sun primes and monogenic power compositional trinomials

A new condition for k-Wall-Sun-Sun primes

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

The Irreducibility and Monogenicity of Power-Compositional Trinomials

Characteristic Polynomials of Simple Ordinary Abelian Varieties over Finite Fields

Monogenic cyclotomic compositions

Monogenic trinomials with non-squarefree discriminant

On Primitive Covering Numbers

Representing Integers as the Sum of Two Squares in the Ring Zn​

Using Lucas Sequences to Generalize a Theorem of Sierpiński

Polynomial Cunningham Chains

A new condition for $k$ -Wall-Sun-Sun primes

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

Representing Integers as the Sum of Two Squares in the Ring $Z_{n}$