Euler's famous prime generating polynomial and the class number of imaginary quadratic fields
type: Item
instance of: scholarly article
title: Euler's famous prime generating polynomial and the class number of imaginary quadratic fields
zbMATH Open document ID: 0663.12003
published in: L'Enseignement Mathématique. 2e Série
publication date: 1988-01-01T00:00:00Z
review text: Are there infinitely many, or only finitely many primes q such that Euler's generalized polynomial \(f_ q(X)=X^ 2+X+q\) has a prime value for \(X=0,1,...,q-2?\) What is the largest possible q? In this connection, the author proves that the following three conditions are equivalent: \((1)\quad q=2,3,5,11,17,41;\) \((2)\quad f_ q(X)\) is a prime for \(X=0,1,2,...,q-2\); \((3)\quad the\) imaginary quadratic field \({\mathbb{Q}}(\sqrt{1-4q})\) has class number one. This lecture also contains some facts from the elementary theory of quadratic fields (units, class number, theorem of Rabinovitch etc.) with a historical excursion.
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11A41
zbMATH DE Number: 4083760
zbMATH Keywords: Euler's generalized polynomial
zbMATH Keywords: quadratic fields
zbMATH Keywords: units
zbMATH Keywords: class number
author: Paulo Ribenboim
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4083760