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Joshua Harrington

Publications and source records attributed to Joshua Harrington.

At least 19 recordsLinked to original sources
arXiv2026

Generalized Wieferich primes and monogenic trinomials

Let $b\ge 2$ be an integer and let $p\ge 3$ 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 $b^{p-1}\equiv 1 \pmod{p^2}$. When $b=2$, $p$ is also known simply as a Wieferich prime. Let $f(x)\in {\mathbb Z}[x]$ be a monic polynomial of degree $N\ge 2$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. Recently, the third author proved that $x^{2p}+2x^p+2$ is monogenic if and only if $p$ is not a Wieferich prime. In this article, we generalize this result to $x^{2n}+bx^n+b$ with certain restrictions on $b\ge 2$ and $n\ge 3$.

math.NT↗
arXiv2026

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

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 multiple of the moduli $2^a 3^b 5^c$. We obtain complete description of all such quadruples when $m=2,3,4,5$, or $6$, except when $m=6$ and $b=c=1$. We also show that if the LCM of the moduli has only $2$, $3$, or $5$ as prime divisors, then $m \leq 9$ and construct a distinct covering system with $m=8$, $a=8$, $b=3$, and $c=2$. When a covering system exists for a quadruple $(m,a,b,c)$ we provide an example. Nonexistence of covering systems is established via integer programming or by using a new estimate on the density of a set covered by a system of congruences.

math.NT↗
arXiv2026

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

Let $f(x)=x^{2p}+ax^p+b^p$, where $p$ is a prime and $a,b\in {\mathbb Z}$ with $ab\ne 0$. If $f(x)$ is irreducible over ${\mathbb Q}$, we say that $f(x)$ is monogenic if $\{1,\theta,\theta^2,\ldots ,\theta^{2p-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we give a characterization of the monogenic trinomials $f(x)$ according to their Galois groups. These results extend prior investigations of the authors.

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arXiv2025

Generalized Sierpi\'{n}ski and Riesel numbers of the form $tb^t+\alpha$

Let $b\geq 2$ be an integer. We call an integer $k$ a $b$-Sierpi\'{n}ski number if $\gcd(k+1,b-1)=1$ and $k\cdot b^n+1$ is composite for all positive integers $n$. We similarly call $k$ a $b$-Riesel number if $\gcd(k-1,b-1)=1$ and $k\cdot b^n-1$ is composite for all positive integers $n$. An integer that is simultaneously $b$-Sierpi\'{n}ski and $b$-Riesel is called a $b$-Brier number. In this article, we show that for any integer $\alpha\neq 0$, there are infinitely many $b$-Sierpi\'{n}ski numbers and infinitely many $b$-Riesel numbers of the form $tb^t+\alpha$. We further show that when $b+1$ is not a power of $2$, there are infinitely $b$-Brier number of this form.

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arXiv2025

Monogenic sextic trinomials $x^6+Ax^3+B$ and their Galois groups

Let $f(x)=x^6+Ax^3+B\in {\mathbb Z}[x]$, with $A\ne 0$, and suppose that $f(x)$ is irreducible over ${\mathbb Q}$. We define $f(x)$ to be {\em monogenic} if $\{1,\theta,\theta^2,\theta^3,\theta^4,\theta^{5}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. For each possible Galois group $G$ of $f(x)$ over ${\mathbb Q}$, we use a theorem of Jakhar, Khanduja and Sangwan to give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. We also investigate when these trinomials generate distinct sextic fields.

math.NT↗
arXiv2025

A further investigation on covering systems with odd moduli

Erd\H{o}s first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in this field is the well-known odd covering problem. In this paper, we investigate a variant of that problem, where one odd integer is permitted to appear multiple times as a modulus in the covering system, while all remaining moduli are distinct odd integers greater than 1.

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arXiv2025

Monogenic Cyclic Polynomials in Recurrence Sequences

Let $f(x)\in {\mathbb Z}[x]$ be an $N$th degree polynomial that is monic and irreducible over ${\mathbb Q}$. We say that $f(x)$ is {\em monogenic} if $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. We say that $f(x)$ is {\em cyclic} if the Galois group of $f(x)$ over ${\mathbb Q}$ is the cyclic group of order $N$. In this article, we investigate the appearance of monogenic cyclic polynomials in certain polynomial recurrence sequences.

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arXiv2025

On Sierpi\'nski and Riesel Repdigits and Repintegers

For positive integers $b\geq 2$, $k<b$, and $t$, we say that an integer $k_b^{(t)}$ is a $b$-repdigit if $k_b^{(t)}$ can be expressed as the digit $k$ repeated $t$ times in base-$b$ representation, i.e., $k_b^{(t)} =k(b^t-1)/(b-1)$. In the case of $k=1$, we say that $1_b^{(t)}$ is a $b$-repunit. In this article, we investigate the existsence of $b$-repdigits and $b$-repunits among the sets of Sierpi\'nski numbers and Riesel numbers. A Sierpi\'nski number is defined as an odd integer $k$ for which $k\cdot 2^n+1$ is composite for all positive integers $n$ and Riesel numbers are similarly defined for the expression $k\cdot 2^n-1$.

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arXiv2024

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

Let $f(x)=x^4+ax^3+d\in {\mathbb Z}[x]$, where $ad\ne 0$. 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 order 12. Assuming that $f(x)$ is monogenic, we give necessary and sufficient conditions involving only $a$ and $d$ to determine the Galois group $G$ of $f(x)$ over ${\mathbb Q}$. In particular, we show that $G=D_4$ if and only if $(a,d)=(\pm 2,2)$, and that $G\not \in \{C_4,C_2\times C_2\}$. Furthermore, we prove that $f(x)$ is monogenic with $G=A_4$ if and only if $a=4k$ and $d=27k^4+1$, where $k\ne 0$ is an integer such that $27k^4+1$ is squarefree. This article extends previous work of the authors on the monogenicity of quartic polynomials and their Galois groups.

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arXiv2024

Monogenic Quartic Polynomials and Their Galois Groups

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren, to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new, and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the current literature. Finally, we devote a separate section to a discussion concerning, what we believe to be, the still-unresolved cyclic case.

math.NT↗
arXiv2024

Paired $(n-1)$-to-$(n-1)$ disjoint path covers in bipartite transposition-like graphs

A paired $k$-to-$k$ disjoint path cover of a graph $G$ is a collection of pairwise disjoint path subgraphs $P_1,P_2,\dotsc,P_k$ such that each $P_i$ has prescribed vertices $s_i$ and $t_i$ as endpoints and the union of $P_1,P_2,\dotsc,P_k$ contains all vertices of $G$. In this paper, we introduce bipartite transposition-like graphs, which are inductively constructed from lower ranked bipartite transposition-like graphs. We show that every rank $n$ bipartite transposition-like graph $G$ admit a paired $(n-1)$-to-$(n-1)$ disjoint path cover for all choices of $S=\{s_1,s_2,\dotsc,s_{n-1}\}$ and $T=\{t_1,t_2,\dotsc,t_{n-1}\}$, provided that $S$ is in one partite set of $G$ and $T$ is in the other.

math.CO↗
arXiv2022

The Irreducibility and Monogenicity of Power-Compositional Trinomials

A polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called \emph{monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,θ,θ^2,\ldots ,θ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(θ)$, where $f(θ)=0$. Define ${\mathcal F}(x):=x^m+Ax^{m-1}+B$. In this article, we determine sets of conditions on $m$, $A$, and $B$, such that the power-compositional trinomial ${\mathcal F}(x^{p^n})$ is monogenic for all integers $n\ge 0$ and a given prime $p$. Furthermore, we prove the actual existence of infinite families of such trinomials ${\mathcal F}(x)$.

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arXiv2022

Residue sums of Dickson polynomials over finite fields

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 sometimes be computed independent of the prime $p$; for example, Gauss showed that the sum over quadratic residues vanishes modulo a prime. In this paper we provide a closed form for the sum over distinct residues in the image of Dickson polynomials of arbitrary degree over finite fields of odd characteristic, and prove a complete characterization of the size of the image set. Our result provides the first non-trivial classification of such a sum for a family of polynomials of unbounded degree.

math.NT↗
arXiv2022

Probabilistic chip-collecting games with modulo winning conditions

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 $b$ chips. If Alice collects $a$ chips, then Bob collects $b$ chips, and vice versa. A player is announced the winner when they have accumulated a number of chips that is a multiple of $n$. In this paper, we settle two conjectures from the literature related to this game.

math.CO↗
arXiv2021

Sum index and difference index of graphs

Let $G$ be a nonempty simple graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V(G)\to\mathbb{Z}$, there are two induced edge labelings, namely $f^+:E(G)\to\mathbb{Z}$ defined by $f^+(uv)=f(u)+f(v)$, and $f^-:E(G)\to\mathbb{Z}$ defined by $f^-(uv)=|f(u)-f(v)|$. The sum index and the difference index are the minimum cardinalities of the ranges of $f^+$ and $f^-$, respectively. We provide upper and lower bounds on the sum index and difference index, and determine the sum index and difference index of various families of graphs. We also provide an interesting conjecture relating the sum index and the difference index of graphs.

math.CO↗
arXiv2021

Covering systems with odd moduli

The concept of a covering system was first introduced by Erd\H{o}s in 1950. Since their introduction, a lot of the research regarding covering systems has focused on the existence of covering systems with certain restrictions on the moduli. Arguably, the most famous open question regarding covering systems is the odd covering problem. In this paper, we explore a variation of the odd covering problem, allowing a single odd prime to appear as a modulus in the covering more than once, while all other moduli are distinct, odd, and greater than $1$. We also consider this variation while further requiring the moduli of the covering system to be square-free.

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