J. Harrington

activity

20142026

most citedSum index and difference index of graphs

6 citations · 19 across the 19 of their papers we have counts for

collaborators

21 papers

math.NT2026

Generalized Wieferich primes and monogenic trinomials

Amy Falk, Joshua Harrington, Lenny Jones

Let $b \geq 2$ be an integer and let $p \geq 3$ be a prime. We say that $p$ is a {\em generalized Wieferich prime base $b$ }, or more succinctly, a {\em base- $b$ Wieferich prime,} if $b…

math.NT2026

Covering systems where the prime divisors of all moduli are only $2$ , $3$ , or $5$

Joshua Harrington, Jonah Klein, Joshua Lowrance +1

We try to find all quadruples of positive integers $(m, a, b, c)$ with $a \geq b \geq c$ such that there exists a distinct covering system with minimum modulus $m$ and least common mu…

math.NT2026

On the monogenicity and Galois groups of $x^{2 p} + a x^{p} + b^{p}$

Joshua Harrington, Lenny Jones

Let $f (x) = x^{2 p} + a x^{p} + b^{p}$ , where $p$ is a prime and $a, b \in Z$ with $ab \neq = 0$ . If $f (x)$ is irreducible over $Q$ , we say that $f (x)$ is monogenic if $\{1,θ,θ^2…

math.NT2025

Generalized Sierpiński and Riesel numbers of the form $t b^{t} + α$

Hailey Evans, Joshua Harrington, Kendall Heiney +1

Let $b \geq 2$ be an integer. We call an integer $k$ a $b$ -Sierpiński number if $g cd (k + 1, b - 1) = 1$ and $k \cdot b^{n} + 1$ is composite for all positive integers $n$ . We similarly call $k…

math.NT2025

Monogenic sextic trinomials $x^{6} + A x^{3} + B$ and their Galois groups

Joshua Harrington, Lenny Jones

Let $f (x) = x^{6} + A x^{3} + B \in Z [x]$ , with $A \neq = 0$ , and suppose that $f (x)$ is irreducible over $Q$ . We define $f (x)$ to be {\em monogenic} if $\{1,θ,θ^2,θ^3,θ^4,θ^{5…

math.NT2025

Monogenic Cyclic Polynomials in Recurrence Sequences

Joshua Harrington, Lenny Jones

Let $f (x) \in Z [x]$ be an $N$ th degree polynomial that is monic and irreducible over $Q$ . We say that $f (x)$ is {\em monogenic} if ${1, θ, θ^{2}, \dots, θ^{N - 1}}$ …

math.NT2024

Monogenic trinomials of the form $x^{4} + a x^{3} + d$ and their Galois groups

Joshua Harrington, Lenny Jones

Let $f (x) = x^{4} + a x^{3} + d \in Z [x]$ , where $a d \neq = 0$ . Let $C_{n}$ denote the cyclic group of order $n$ , $D_{4}$ the dihedral group of order 8, and $A_{4}$ the alternating group of or…

math.NT2024★ 6 cited

Monogenic Quartic Polynomials and Their Galois Groups

Joshua Harrington, Lenny Jones

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

math.NT2023

Arithmetic progressions of integers that are relatively prime to their digital sums

Ryan Blau, Joshua Harrington, Sarah Lohrey +2

For an integer $b \geq 2$ , we call a positive integer $b$ -anti-Niven if it is relatively prime to the sum of the digits in its base- $b$ representation. In this article, we investiga…

math.NT2023★ 2 cited

Every arithmetic progression contains infinitely many $b$ -Niven numbers

Joshua Harrington, Matthew Litman, Tony W. H. Wong

For an integer $b \geq 2$ , a positive integer is called a $b$ -Niven number if it is a multiple of the sum of the digits in its base- $b$ representation. In this article, we show that…

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.CO2022

Probabilistic chip-collecting games with modulo winning conditions

Joshua Harrington, Xuwen Hua, Xufei Liu +3

Let $a$ , $b$ , and $n$ be integers with $0 < a < b < n$ . In a certain two-player probabilistic chip-collecting game, Alice tosses a coin to determine whether she collects $a$ chips or $b$ …

math.NT2021★ 1 cited

Covering systems with odd moduli

Joshua Harrington, Yewen Sun, Wing Hong Tony Wong

The concept of a covering system was first introduced by Erdős in 1950. Since their introduction, a lot of the research regarding covering systems has focused on the existence of c…

math.NT2021

Residue sums of Dickson polynomials over finite fields

Thomas Brazelton, Joshua Harrington, Matthew Litman +1

Given a polynomial with integral coefficients, one can inquire about the possible residues it can take in its image modulo a prime $p$ . The sum over the distinct residues can somet…

math.NT2020

On binomial coefficients associated with Sierpiński and Riesel numbers

Ashley Armbruster, Grace Barger, Sofya Bykova +5

In this paper, we investigate the existence of Sierpiński numbers and Riesel numbers as binomial coefficients. We show that for any odd positive integer $r$ , there exist infinitely…

math.NT2020

On the properties of fibotomic polynomials

Cameron Byer, Tyler Dvorachek, Emily Eckard +3

Define the $n$ -th fibotomic polynomial to be the product of the monic irredicible factors of the $n$ -th Fibonacci polynomial which are not factors of any Fibonacci polynomial of sm…

math.CO2020★ 6 cited

Sum index and difference index of graphs

Joshua Harrington, Eugene Henninger-Voss, Kedjar Karhadkar +2

Let $G$ be a nonempty simple graph with a vertex set $V (G)$ and an edge set $E (G)$ . For every injective vertex labeling $f : V (G) \to Z$ , there are two induced edge labelings,…

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.NT2018★ 1 cited

An Investigation on Partitions with Equal Products

Byungchul Cha, Adam Claman, Joshua Harrington +6

An ordered triple $(s, p, n)$ is called admissible if there exist two different multisets $X = {x_{1}, x_{2}, \dots, x_{n}}$ and $Y = {y_{1}, y_{2}, \dots, y_{n}}$ such that $X$ and $Y$ share the sa…

math.CO2018★ 1 cited

On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points

Theresa Baren, Michael Cory, Mia Friedberg +6

A permutation graph $G_{π}$ is a simple graph with vertices corresponding to the elements of $π$ and an edge between $i$ and $j$ when $i$ and $j$ are inverted in $π$ . A set of vertic…

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…

Generalized Wieferich primes and monogenic trinomials

Covering systems where the prime divisors of all moduli are only 2, 3, or 5

On the monogenicity and Galois groups of x2p+axp+bp

Generalized Sierpiński and Riesel numbers of the form tbt+α

Monogenic sextic trinomials x6+Ax3+B and their Galois groups

Monogenic Cyclic Polynomials in Recurrence Sequences

Monogenic trinomials of the form x4+ax3+d and their Galois groups

Monogenic Quartic Polynomials and Their Galois Groups

Arithmetic progressions of integers that are relatively prime to their digital sums

Every arithmetic progression contains infinitely many b-Niven numbers

The Irreducibility and Monogenicity of Power-Compositional Trinomials

Probabilistic chip-collecting games with modulo winning conditions

Covering systems with odd moduli

Residue sums of Dickson polynomials over finite fields

On binomial coefficients associated with Sierpiński and Riesel numbers

On the properties of fibotomic polynomials

Sum index and difference index of graphs

Monogenic cyclotomic compositions

An Investigation on Partitions with Equal Products

On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points

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

Covering systems where the prime divisors of all moduli are only $2$ , $3$ , or $5$

On the monogenicity and Galois groups of $x^{2 p} + a x^{p} + b^{p}$

Generalized Sierpiński and Riesel numbers of the form $t b^{t} + α$

Monogenic sextic trinomials $x^{6} + A x^{3} + B$ and their Galois groups

Monogenic trinomials of the form $x^{4} + a x^{3} + d$ and their Galois groups

Every arithmetic progression contains infinitely many $b$ -Niven numbers

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