scholarly journals Artin's Primitive Root Conjecture – A Survey

Integers ◽  
2012 ◽  
Vol 12 (6) ◽  
Author(s):  
Pieter Moree
Keyword(s):  
Number Theory ◽  
Primitive Root ◽  
Number Fields ◽  
Elementary Number Theory ◽  
Open Problems ◽  
General Audience ◽  
Elementary Number ◽  
Artin's Conjecture ◽  
Multiplicative Order ◽  
Artin Primitive Root Conjecture

Abstract.One of the first concepts one meets in elementary number theory is that of the multiplicative order. We give a survey of the literature on this topic emphasizing the Artin primitive root conjecture (1927). The first part of the survey is intended for a rather general audience and rather colloquial, whereas the second part is intended for number theorists and ends with several open problems. The contributions in the survey on `elliptic Artin' are due to Alina Cojocaru. Wojciec Gajda wrote a section on `Artin for K-theory of number fields,' and Hester Graves (together with me) on `Artin's conjecture and Euclidean domains.'

Elementary Number Theory

2016 ◽  
pp. 1-32
Author(s):  
Gary L. Mullen ◽  
James A. Sellers
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number

Some Elementary Number Theory

2019 ◽  
pp. 239-244
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Result Section ◽  
Structural Result ◽  
Elementary Number

This chapter proves some number-theoretic results about the sequences defined in Chapter 23. It proceeds as follows. Section 24.2 proves Lemma 24.1, a multipart structural result. Section 24.3 takes care of several number-theoretic details left over from Section 23.6 and Section 23.7.

Infinite Orbits Revisited

2019 ◽  
pp. 227-238
Author(s):  
Richard Evan Schwartz
Keyword(s):  
Number Theory ◽  
Elementary Number Theory ◽  
Elementary Number ◽  
Model Section

This is the first of four chapters giving a self-contained proof of Theorem 0.7. Section 23.2 describes a sequence of even rationals {pn/qn} that converges to A. Section 23.3 states the two main technical results, the Box Theorem and the Copy Theorem. Section 23.4 shows how to choose a sequence {cn}. Section 23.5 states three auxiliary results about arc copying in the plaid model. Section 23.6 deduces the Box Theorem from one of these auxiliary lemmas. Section 23.7 deduces the Copy Theorem from the auxiliary lemmas and some elementary number theory. Thus, after this chapter ends, the only remaining task is to prove the auxiliary copy lemmas and prove a few lemmas in elementary number theory.

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