Arithmetic consequences of Jacobi's two-squares theorem
type: Item
instance of: scholarly article
title: Arithmetic consequences of Jacobi's two-squares theorem
zbMATH Open document ID: 1003.11011
DOI: 10.1023/A:1009877906185
author: Michael D. Hirschhorn
published in: The Ramanujan Journal
publication date: 2000-06-05T00:00:00Z
review text: Let \(r_2(n)\) denote the number of representations of \(n\) as a sum of two squares. Jacobi's formula for \(r_2(n)\) easily implies: \[ \begin{alignedat}{2} r_2(2n)&= r_2(n), &&\\ r_2(pn)&= 2r_2(n)- r_2 \bigl( \tfrac{n}{p} \bigr) &\quad\text{for }&p\in\mathbb{P},\;p\equiv 1\bmod 4,\\ r_2(pn)&= r_2 \bigl( \tfrac{n}{p} \bigr) &\quad\text{for }&p\in\mathbb{P},\;p\equiv 3\bmod 4.\end{alignedat} \] These formulas are proved directly via generating functions. They easily imply Jacobi's two-square theorem.
Mathematics Subject Classification ID: 11E25
zbMATH DE Number: 1455278
zbMATH Keywords: sums of squares
zbMATH Keywords: Jacobi's two-squares theorem
reviewed by: Meinhard Peters
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1455278