Carmichael's ``Empirical Theorem''
type: Item
instance of: scholarly article
title: Carmichael's ``Empirical Theorem''
zbMATH Open document ID: 0585.10003
DOI: 10.1007/BF03026838
author: Stan Wagon
published in: The Mathematical Intelligencer
publication date: 1986-01-01T00:00:00Z
review text: If \(n\) is a natural number and \(\varphi\) is Euler's totient function, let \(N(n)\) denote the number of solutions of the equation \(\varphi (x)=n\). More than sixty years ago R. D. Carmichael conjectured that for every \(n\) either \(N(n)=0\) or \(N(n)\geq 2\). This conjecture is equivalent to the assertion that if \(X=\{x: N(\varphi (x))=1\}\) then \(X\) is empty. The present paper discusses the history of Carmichael's conjecture and the progress which has been made in the study of the elements of \(X\). Particular attention is paid to the result obtained by \textit{P. Masai} and \textit{A. Valette} [Boll. Unione Mat. Ital., IV. Ser., A 1, 313--316 (1982; Zbl 0484.10004)] that if \(x\in X\) then \(x>10^{10 000}\). The author points out that a very significant improvement in this lower bound for \(X\) could almost certainly be obtained by making only a minor adjustment in the Masai-Valette method.
reviewed by: Peter Hagis jun.
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 3937236
zbMATH Keywords: large lower bound
zbMATH Keywords: Carmichael-Klee theorem
zbMATH Keywords: Euler phi-function
zbMATH Keywords: Euler's totient function
zbMATH Keywords: history of Carmichael's conjecture
zbMATH Keywords: Masai-Valette method
MaRDI profile type: Publication
cites work: On a conjecture of Carmichael
cites work: Q3943022
cites work: On Carmichael's Conjecture
cites work: Recent developments in primality testing
cites work: The Pseudoprimes to 25 ⋅10 9
description: scientific article; zbMATH DE number 3937236