Jeremy A. Rouse

activity

20072023

most citedGalois theory of iterated endomorphisms

44 citations · 123 across the 26 of their papers we have counts for

collaborators

31 papers

math.NT2023

Densities of integer sets represented by quadratic forms

Pete L. Clark, Paul Pollack, Jeremy Rouse +1

Let $f (t_{1}, \dots, t_{n})$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_{f} (X)$ , the number of integers of absolute value up to $X$ …

math.NT2022

Quaternary quadratic forms with prime discriminant

Jeremy Rouse, Katherine Thompson

Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$ .…

math.NT2022

Rational points on $x^{3} + x^{2} y^{2} + y^{3} = k$

Xiaoan Lang, Jeremy Rouse

We study the problem of determining, given an integer $k$ , the rational solutions to $C_{k} : x^{3} z + x^{2} y^{2} + y^{3} z = k z^{4}$ . For $k \neq = 0$ , the curve $C_{k}$ has genus $3…

math.NT2021★ 29 cited

$ℓ$ -adic images of Galois for elliptic curves over $Q$

Jeremy Rouse, Andrew V. Sutherland, David Zureick-Brown

We discuss the $ℓ$ -adic case of Mazur's "Program B" over $Q$ , the problem of classifying the possible images of $ℓ$ -adic Galois representations attached to elliptic…

math.NT2021

Integers that are sums of two rational sixth powers

Alexis Newton, Jeremy Rouse

We prove that $164634913$ is the smallest positive integer that is a sum of two rational sixth powers but not a sum of two integer sixth powers. If $C_{k}$ is the curve $x^{6} + y^…

math.NT2020★ 3 cited

Odd degree isolated points on $X_{1} (N)$ with rational $j$ -invariant

Abbey Bourdon, David R. Gill, Jeremy Rouse +1

Let $C$ be a curve defined over a number field $k$ . We say a closed point $x \in C$ of degree $d$ is isolated if it does not belong to an infinite family of degree $d$ points parame…

math.NT2019

The number of representations of $n$ as a growing number of squares

John Holley-Reid, Jeremy Rouse

Let $r_{k} (n)$ denote the number of representations of the integer $n$ as a sum of $k$ squares. In this paper, we give an asymptotic for $r_{k} (n)$ when $n$ grows linearly with $k$ …

math.NT2019

Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

Michael Cerchia, Jeremy Rouse

Let $ℓ$ be a prime number and let $F$ be a number field and $E / F$ a non-CM elliptic curve with a point $α \in E (F)$ of infinite order. Attached to the pair $(E, α)$ is the $ℓ$ -…

math.NT2018

$p$ -adic quotient sets II: quadratic forms

Christopher Donnay, Stephan Ramon Garcia, Jeremy Rouse

For $A \subseteq {1, 2, \dots}$ , we consider $R (A) = {a / a^{'} : a, a^{'} \in A}$ . If $A$ is the set of nonzero values assumed by a quadratic form, when is $R (A)$ dense in the $p$ -adic…

math.NT2018

The density of odd order reductions for elliptic curves with a rational point of order 2

Ke Liang, Jeremy Rouse

Suppose that $E / Q$ is an elliptic curve with a rational point $T$ of order $2$ and $α \in E (Q)$ is a point of infinite order. We consider the problem of determinin…

math.NT2018

Integers represented by positive-definite quadratic forms and Petersson inner products

Jeremy Rouse

Let $Q$ be a positive-definite quaternary quadratic form with integer coefficients. We study the problem of giving bounds on the largest positive integer $n$ that is locally repres…

math.NT2017

An Elliptic Curve Analogue to the Fermat Numbers

Skye Binegar, Randy Dominick, Meagan Kenney +2

The Fermat numbers have many notable properties, including order universality, coprimality, and definition by a recurrence relation. We use arbitrary elliptic curves and rational p…

math.NT2016

Quadratic forms representing all integers coprime to 3

Justin DeBenedetto, Jeremy Rouse

Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers…

math.NT2016

When is $a^{n} + 1$ the sum of two squares?

Greg Dresden, Kylie Hess, Saimon Islam +5

Using Fermat's two squares theorem and properties of cyclotomic polynomials, we prove assertions about when numbers of the form $a^{n} + 1$ can be expressed as the sum of two integer…

math.NT2016

Sum of two repdigits a square

Bart Goddard, Jeremy Rouse

A \emph{repdigit} is a natural number greater than 10 which has all of its base-10 digits the same. In this paper we find all examples of two repdigits adding to a square. The proo…

math.NT2015

Trinomials defining quintic number fields

Jesse Patsolic, Jeremy Rouse

Given a quintic number field $K / Q$ , we study the set of irreducible trinomials, polynomials of the form $x^{5} + a x + b$ , that have a root in $K$ . We show that there is a…

math.NT2015

The density of primes dividing a particular non-linear recurrence sequence

Alexi Block Gorman, Tyler Genao, Heesu Hwang +3

Define the sequence ${b_{n}}$ by $b_{0} = 1, b_{1} = 1, b_{2} = 2, b_{3} = 1$ , and $$b_n=\begin{cases} \frac{b_{n-1}b_{n-3}-b_{n-2}^2}{b_{n-4}}&\textrm{if}~ n\not\equiv 0\pmod 3, \frac{b_{n-1}b_{n-3…

math.NT2015

The density of primes dividing a term in the Somos-5 sequence

Bryant Davis, Rebecca Kotsonis, Jeremy Rouse

The Somos-5 sequence is defined by $a_{0} = a_{1} = a_{2} = a_{3} = a_{4} = 1$ and $a_{m} = \frac{a _{m - 1} a _{m - 4} + a _{m - 2} a _{m - 3}}{a _{m - 5}}$ for $m \geq 5$ . We relate the arithme…

math.NT2015

Three consecutive almost squares

Jeremy Rouse, Yilin Yang

Given a positive integer $n$ , we let $sfp (n)$ denote the squarefree part of $n$ . We determine all positive integers $n$ for which $\max \{ {\rm sfp}(n), {\rm sfp}(n+1), {\rm…

math.NT2014

Elliptic curves over $Q$ and 2-adic images of Galois

Jeremy Rouse, David Zureick-Brown

We give a classification of all possible $2$ -adic images of Galois representations associated to elliptic curves over $Q$ . To this end, we compute the 'arithmetically maxi…

math.NT2013★ 3 cited

On spaces of modular forms spanned by eta-quotients

Jeremy Rouse, John J. Webb

An eta-quotient of level $N$ is a modular form of the shape $f (z) = \prod_{δ ∣ N} η (δ z)^{r_{δ}}$ . We study the problem of determining levels $N$ for which the graded ring of holomorph…

math.NT2013★ 18 cited

The Explicit Sato-Tate Conjecture and Densities Pertaining to Lehmer-Type Questions

Jeremy Rouse, Jesse Thorner

Let $f (z) = \sum_{n = 1}^{\infty} a (n) q^{n} \in S_{k}^{new} (Γ_{0} (N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$ , define $θ_p\in…

math.NT2012★ 1 cited

Divisibility properties of the Fibonacci entry point

Paul Cubre, Jeremy Rouse

For a prime $p$ , let $Z (p)$ be the smallest positive integer $n$ so that $p$ divides $F_{n}$ , the $n$ th term in the Fibonacci sequence. Paul Bruckman and Peter Anderson conjectured…

math.NT2012★ 3 cited

Solutions of the cubic Fermat equation in quadratic fields

Marvin Jones, Jeremy Rouse

We give necessary and sufficient conditions on a squarefree integer $d$ for there to be non-trivial solutions to $x^{3} + y^{3} = z^{3}$ in $\Q(\sqrt{d})$, conditional on the Birch…

Densities of integer sets represented by quadratic forms

Quaternary quadratic forms with prime discriminant

Rational points on x3+x2y2+y3=k

ℓ-adic images of Galois for elliptic curves over Q

Integers that are sums of two rational sixth powers

Odd degree isolated points on X1​(N) with rational j-invariant

The number of representations of n as a growing number of squares

Uniform bounds on the image of the arboreal Galois representations attached to non-CM elliptic curves

p-adic quotient sets II: quadratic forms

The density of odd order reductions for elliptic curves with a rational point of order 2

Integers represented by positive-definite quadratic forms and Petersson inner products

An Elliptic Curve Analogue to the Fermat Numbers

Quadratic forms representing all integers coprime to 3

When is an+1 the sum of two squares?

Sum of two repdigits a square

Trinomials defining quintic number fields

The density of primes dividing a particular non-linear recurrence sequence

The density of primes dividing a term in the Somos-5 sequence

Three consecutive almost squares

Elliptic curves over Q and 2-adic images of Galois

On spaces of modular forms spanned by eta-quotients

The Explicit Sato-Tate Conjecture and Densities Pertaining to Lehmer-Type Questions

Divisibility properties of the Fibonacci entry point

Solutions of the cubic Fermat equation in quadratic fields

Rational points on $x^{3} + x^{2} y^{2} + y^{3} = k$

$ℓ$ -adic images of Galois for elliptic curves over $Q$

Odd degree isolated points on $X_{1} (N)$ with rational $j$ -invariant

The number of representations of $n$ as a growing number of squares

$p$ -adic quotient sets II: quadratic forms

When is $a^{n} + 1$ the sum of two squares?

Elliptic curves over $Q$ and 2-adic images of Galois