Corrigendum The Generalized Artin Primitive Root Conjecture (EUROPEAN J. OF PURE AND APPLIED MATH. Vol. 11, No. 1, 2018, 23-34)

The subset of integers $\mathcal{N}_2= \{ n\in \mathbb{N}:\text{ord}_n(2)=\lambda(n) \}$ in page 24, \cite{CN18}, should be \\ $$\label{eq2-40} \mathcal{N}_u =\left\{ n\in \mathbb{N}:\ord_n(u)=\lambda(n) \text{ and } p \mid n \Rightarrow \ord_p(u)=p-1, \ord_{p^2}(u)=p(p-1) \right\} $$ where $u \ne \pm 1, v^2$.In addition, Lemma 3.4 was corrected. These changes do not affect the main result. The proof of Theorem 1.1 remains the same. The new version of the paper is available at arXiv:1504.00843v9.

On Self-Dual Four Circulant Codes

Four circulant codes form a special class of [Formula: see text]-generator, index [Formula: see text], quasi-cyclic codes. Under some conditions on their generator matrix they can be shown to be self-dual. Artin primitive root conjecture shows the existence of an infinite subclass of these codes satisfying a modified Gilbert–Varshamov bound.

The Generalized Artin Primitive Root Conjecture

Asymptotic formulas for the number of integers with the primitive root 2, and the generalized Artin conjecture for subsets of composite integers with fixed admissible primitive roots \(u\neq \pm 1,v^2\), are presented here.

Artin's Primitive Root Conjecture – A Survey

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.'