Sums of Quaternion Squares and a Theorem of Watson

Tim Banks, Spencer Hamblen, Tim Sherwin, Sal Wright

发表信息：: Involve 14 (2021) 783-792
DOI：: 10.2140/involve.2021.14.783

英文摘要：

We use a representability theorem of G. L. Watson to examine sums of squares in Quaternion rings with integer coefficients. This allows us to determine a large family of such rings where every element expressible as the sum of squares can be written as the sum of 3 squares.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Sums of Cubes in Quaternion Rings

We investigate a version of Waring's Problem over quaternion rings, focusing on cubes in quaternion rings with integer coefficients. We determine the global upper and lower bounds for the number of cubes necessary to represent all such quaternions.

2019-10-06math.NT

Sums of squares in Quaternion rings

Lagrange's Four Squares Theorem states that any positive integer can be expressed as the sum of four integer squares. We investigate the analogous question over Quaternion rings, focusing on squares of elements of Quaternion rings with integer coefficients. We determine the minimum necessary number of squares for infinitely many Quaternion rings, and give global upper and lower bounds.

2016-10-23math.NT

Restricted sums of four integral squares

We give a simple quaternionic proof of a recent result of Goldmakher and Pollack on restricted sums of four integral squares.

2019-03-09math.NT

Numbers represented by restricted sums of four squares

In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$ where $x,y,z,t$ are integers and $x+y+2z+2t$ is a square or a cube.

2021-05-28math.NT

Finding Sums of Four Squares via Complex Continued Fractions

The problem of representing a given positive integer as a sum of four squares of integers has been widely concerned for a long time, and for a given positive odd $n$ one can find a representation by doing arithmetic in a maximal order of quaternion algebra once a pair of (positive) integers $x,y$ with $x^2+y^2\equiv-1\mod n$ is given. In this paper, we introduce a new method to find a representation of odd integer $w$ given $x,y$ satisfying the above requirement. This method can avoid the complicated non-commutative structure in quaternion algebra, which is similar to the one we use to obtain …

2022-05-02math.NT

Sums of squares and orthogonal integral vectors

Two vectors in $\BZ^3$ are called \emph{twins} if they are orthogonal and have the same length. The paper describes twin pairs using cubic lattices, and counts the number of twin pairs with a given length. Integers $M$ with the property that each integral vector with length $\sqrt{M}$ has a twin are called twin-complete. They are completely characterized modulo a famous conjecture in number theory. The main tool is the decomposition theory of Hurwitz integral quaternions. Throughout the paper we made a concerted effort to keep the exposition as elementary as possible.

2011-08-10math.NT

更多相关论文收起更多论文

Baruah--Gogoi关于奇数和偶数上划线部分之和的猜想的一个证明

A proof of the Baruah--Gogoi conjecture on sums of odd and even overlined parts

Shanghai University of International Business and Economics（上海对外经贸大学）；Suzhou University of Science and Technology（苏州科技大学）

本文用初等q-级数方法（结合Watson五重积恒等式特化、除数级数参数化与逐项对数微分）证明了Baruah-Gogoi关于超分拆奇偶上划线部分和统计量模7同余的猜想。

2026-09-24math.NT

循环除代数中极大序上的矩阵作为平方和与立方和

Matrices over maximal orders in cyclic division algebras over Q as sums of squares and cubes

本文研究有理数域上奇素数次循环除代数中极大序上的矩阵，证明在特定条件下，任意阶数大于等于2的矩阵可表示为平方和与立方和，推广了已有结果。

2025-10-15math.NT

Counting invertible sums of squares modulo $n$ and a new generalization of Euler totient function

In this paper we introduce and study a family $Φ_k$ of arithmetic functions generalizing Euler's totient function. These functions are given by the number of solutions to the equation $\gcd(x_1^2+\ldots +x_k^2, n)=1$ with $x_1,\ldots,x_k \in {\mathbb{Z}}/n{\mathbb{Z}}$ which, for $k=2,4$ and $8$ coincide, respectively, with the number of units in the rings of Gaussian integers, quaternions and octonions over ${\mathbb{Z}}/n{\mathbb{Z}}$. We prove that $Φ_k$ is multiplicative for every $k$, we obtain an explicit formula for $Φ_k(n)$ in terms of the prime-power decomposition of $n$ and derive an…

2014-06-25math.NT