H. Graves

activity

20092026

most citedEuclidean Ideals in Quadratic Imaginary Fields

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

collaborators

12 papers

math.NT2026

Sizes of Pre-Images of the Minimal Euclidean Function on the Gaussian Integers

Hester Graves

In 2023, the author presented the first computable minimal Euclidean function for a non-trivial number field. Along with a formula for $ϕ_{Z [i]}$ , the minimal Euclidean fu…

math.NT2025

The abc conjecture implies infinitely many non-Wieferich places for fixed bases in number fields

Hester Graves, Benjamin Weiss

Silverman showed that, assuming the $ab c$ conjecture, there are $≫ lo g x$ non-Wieferich primes base $a$ less than $x$ \cite{silverman}, for all non-zero $a$ . This inspired Grave…

math.NT2025

A Division Algorithm for the Gaussian Integers' Minimal Euclidean Function

Hester Graves

The usual division algorithms on $Z$ and $Z [i]$ measure the size of remainders using the norm function. These rings are Euclidean with respect to several function…

math.NT2025

Primes of the Form $m^{2} + 1$ and Goldbach's `Other Other' Conjecture

Jon Grantham, Hester Graves

We compute all primes up to $6.25 \times 1 0^{28}$ of the form $m^{2} + 1$ . Calculations using this list verify, up to our bound, a less famous conjecture of Goldbach. We introduce `Gold…

math.NT2022

An Elementary Proof of the Minimal Euclidean Function on the Gaussian Integers

Hester Graves

Every Euclidean domain $R$ has a minimal Euclidean function, $ϕ_{R}$ . A companion paper \cite{Graves} introduced a formula to compute $ϕ_{Z [i]}$ . It is the first formula for…

math.NT2021

Only finitely many $s$ -Cullen numbers are repunits for a fixed $s \geq 2$

Michael Filaseta, Jon Grantham, Hester Graves

We show that for any integer $s \geq 2$ , there are only finitely many $s$ -Cullen numbers that are repunits. More precisely, for fixed $s \geq 2$ , there are only finitely many intege…

math.NT2021

The Minimal Euclidean Function on the Gaussian Integers

Hester Graves

In 1949, Motzkin proved that every Euclidean domain $R$ has a minimal Euclidean function, $ϕ_{R}$ . He showed that when $R = Z$ , the minimal function is $ϕ_{\mathbb{Z}}(x) =…

math.NT2020★ 1 cited

The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits

Jon Grantham, Hester Graves

Assuming the abc conjecture with $ε = 1/6$ , we use elementary methods to show that only finitely many $s$ -Cullen numbers are repunits, aside from two known infinite families. More pr…

math.NT2019

Brazilian Primes Which Are Also Sophie Germain Primes

Jon Grantham, Hester Graves

We disprove a conjecture of Schott that no Brazilian primes are Sophie Germain primes. We enumerate all counterexamples up to $1 0^{46}$ .

math.NT2018

The Minimal Euclidean Function on the Gaussian Integers

Hester Graves

The Minimal Euclidean Function on the Gaussian Integers

math.NT2010

Growth Results and Euclidean Ideals

Hester Graves

Lenstra's concept of Euclidean ideals generalizes the Euclidean algorithm; a domain with a Euclidean ideal has cyclic class group, while a domain with a Euclidean algorithm has tri…

math.NT2009★ 2 cited

Euclidean Ideals in Quadratic Imaginary Fields

Hester Graves, Nick Ramsey

We classify all quadratic imaginary number fields that have a Euclidean ideal class. There are seven of them, they are of class number at most two, and in each case the unique clas…

Sizes of Pre-Images of the Minimal Euclidean Function on the Gaussian Integers

The abc conjecture implies infinitely many non-Wieferich places for fixed bases in number fields

A Division Algorithm for the Gaussian Integers' Minimal Euclidean Function

Primes of the Form m2+1 and Goldbach's `Other Other' Conjecture

An Elementary Proof of the Minimal Euclidean Function on the Gaussian Integers

Only finitely many s-Cullen numbers are repunits for a fixed s≥2

The Minimal Euclidean Function on the Gaussian Integers

The abc Conjecture Implies That Only Finitely Many s-Cullen Numbers Are Repunits

Brazilian Primes Which Are Also Sophie Germain Primes

The Minimal Euclidean Function on the Gaussian Integers

Growth Results and Euclidean Ideals

Euclidean Ideals in Quadratic Imaginary Fields

Primes of the Form $m^{2} + 1$ and Goldbach's `Other Other' Conjecture

Only finitely many $s$ -Cullen numbers are repunits for a fixed $s \geq 2$