An elementary evaluation of the Jacobsthal sum
type: Item
instance of: scholarly article
title: An elementary evaluation of the Jacobsthal sum
zbMATH Open document ID: 0577.10035
DOI: 10.1016/0022-314X(86)90028-4
author: M. G. Monzingo
published in: Journal of Number Theory
publication date: 1986-01-01T00:00:00Z
review text: The paper contains a very interesting and elementary new proof of the theorem of Jacobsthal that if \(p\equiv 1 (4)\) is prime so that \(p=a^ 2+b^ 2\), \(a\equiv -1 (4)\) (a unique) and if \((\frac{r}{p})=1\), say \(r\equiv u^ 2 (mod p)\), then \[ S_ r=\sum^{p- 1}_{x=1}(\frac{x(x^ 2+r)}{p})=(\frac{u}{p}) 2a. \] The machinery used is the decomposition of a reduced residue system 1,2,...,p-1 into 'cells' and then defining singletons, right and left end points and interior points, their number being denoted by s, e and i, respectively. The author then proves easy results connecting the quantities s,e,i amongst themselves and with \(S_ r\) and the theorem then emerges rather elegantly ! It would be nice to see some more applications of the quantities s,e,i.
Mathematics Subject Classification ID: 11L10
zbMATH DE Number: 3922771
zbMATH Keywords: Jacobsthal sum
zbMATH Keywords: Legendre symbol
zbMATH Keywords: character sums
zbMATH Keywords: elementary proof
zbMATH Keywords: reduced residue system
zbMATH Keywords: singletons
zbMATH Keywords: end points
zbMATH Keywords: interior points
reviewed by: A. R. Rajwade
MaRDI profile type: Publication
cites work: Q3666917
cites work: Theorems analogous to Jacobsthal's theorem
cites work: Cyclotomy and Jacobsthal Sums
cites work: Sums of Gauss, Jacobi, and Jacobsthal
full work available at URL: 0022-314x(86)90028-4
OpenAlex ID: W1987700698
description: scientific article; zbMATH DE number 3922771