Award Date

May 2025

Degree Type

Thesis

Degree Name

Master of Science (MS)

Department

Mathematical Sciences

First Committee Member

Peter Shiue

Second Committee Member

Ebrahim Salehi

Third Committee Member

Zhijian Wu

Fourth Committee Member

Rama Venkat

Number of Pages

52

Abstract

Euler’s totient function, φ(n), is the arithmetic function defined as the number of positive integers less than or equal to n that are relatively prime to n. In his 1922 paper [3], Professor R. D. Carmichael conjectured that for each positive integer n, there exists at least one positive integer m̸ = n such that φ(m) = φ(n).In this thesis, we consider some relevant literature and explore Carmichael’s totient conjecture for particular values of φ(n) = k. Our main consideration will be the set X_k = {n ∈ N : φ(n) = k}. In identifying X_k for k = 2^t, 2p^s, 2^2p, and 2pq, we find that Carmichael’s conjecture holds for those select cases, provide an algorithm, and some related results. The conjecture remains an open problem in number theory [10, 24].

Keywords

Carmichael conjecture; Euler totient; Fermat prime; Germain prime

Disciplines

Mathematics

Degree Grantor

University of Nevada, Las Vegas

Language

English

Rights

IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/

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Mathematics Commons

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