J. Harrington

activity

20242026

collaborators

7 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,Î¸,Î¸…

math.NT2025

Generalized SierpiÅski and Riesel numbers of the form $t b^{t} + I^\pm$

Hailey Evans, Joshua Harrington, Kendall Heiney +1

Let $b \geq 2$ be an integer. We call an integer $k$ a $b$ -SierpiÅski 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 $…

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,Î¸,Î¸^2,Î¸^3,Î¸^4,…

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 ${1, θ, θ^{2}, \dots, θ^{N - 1}}$ …

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 SierpiÅski and Riesel numbers of the form tbt+I^±

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

Monogenic Cyclic Polynomials in Recurrence Sequences

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 SierpiÅski and Riesel numbers of the form $t b^{t} + I^\pm$

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