Two theorems of Glaisher and Kaplansky
type: Item
instance of: scholarly article
title: Two theorems of Glaisher and Kaplansky
zbMATH Open document ID: 1233.11041
author: David Brink
published in: Functiones et Approximatio. Commentarii Mathematici
publication date: 2010-01-07T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.facm/1261157807
review text: The author presents a proof of the following theorem of \textit{I. Kaplansky} [Proc. Am. Math. Soc. 131, No. 7, 2299--2300 (2003; Zbl 1055.11026)]: A prime \(p \equiv 1\pmod{16}\) is representable by both or neither of \(x^2+32y^2\) and \(x^2+64y^2\), whereas a prime \(p \equiv 9\pmod{16}\) is representable by exactly one of these forms. The proof described here is based on the arithmetic of the eighth cyclotomic field. Also, the relationship of this result to a theorem of \textit{J. W. L. Gaisher} [Q. J. Math. 34, 178--204 (1902; JFM 34.0243.01)] is pointed out.
reviewed by: Andrew G. Earnest
Mathematics Subject Classification ID: 11E25
Mathematics Subject Classification ID: 11E16
Mathematics Subject Classification ID: 11E41
zbMATH DE Number: 5654316
zbMATH Keywords: binary quadratic forms
zbMATH Keywords: class numbers
zbMATH Keywords: representation of primes
MaRDI profile type: Publication
OpenAlex ID: W2085196406
cites work: Note on primes of type x2 + 32y2, class number, and residuacity.
cites work: Five peculiar theorems on simultaneous representation of primes by quadratic forms
cites work: Über die Klassenzahl des Körpers \(\text{P}\left(\sqrt{-p}\right)\) mit einer Primzahl \(p\equiv1\pmod{2^ 3}\)
cites work: Über die Klassenzahl des Körpers \(P(\sqrt{-2p})\) mit einer Primzahl \(p\ne 2\)
cites work: The forms $x+32y^2$ and $x+64y^2$
DOI: 10.7169/FACM/1261157807
description: scientific article; zbMATH DE number 5654316