Note on Artin's Conjecture on Primitive Roots

E. Artin conjectured that any integer $a > 1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $ p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R. Murty and S. Srinivasan \cite{Murty-Srinivasan} showed that if $\displaystyle \sum_{p < x} \frac 1 {f_a(p)} = O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the Murty-Srinivasan condition to sums involving the cyclotomic periods from the subfields of $\mathbb Q(e^{2\pi i /p})$ corresponding to the subgroups $<a> \subseteq \mathbb F_p^*.$

Explicit upper bound on the least primitive root modulo p2

In this paper, we give an explicit upper bound on [Formula: see text], the least primitive root modulo [Formula: see text]. Since a primitive root modulo [Formula: see text] is not primitive modulo [Formula: see text] if and only if it belongs to the set of integers less than [Formula: see text] which are [Formula: see text]th power residues modulo [Formula: see text], we seek the bounds for [Formula: see text] and [Formula: see text] to find [Formula: see text] which satisfies [Formula: see text], where, [Formula: see text] denotes the number of primitive roots modulo [Formula: see text] not exceeding [Formula: see text], and [Formula: see text] denotes the number of [Formula: see text]th powers modulo [Formula: see text] not exceeding [Formula: see text]. The method we mainly use is to estimate the character sums contained in the expressions of the [Formula: see text] and [Formula: see text] above. Finally, we show that [Formula: see text] for all primes [Formula: see text]. This improves the recent result of Kerr et al.

CUBES IN FINITE FIELDS AND RELATED PERMUTATIONS

Let $p=3n+1$ be a prime with $n\in \mathbb {N}=\{0,1,2,\ldots \}$ and let $g\in \mathbb {Z}$ be a primitive root modulo p. Let $0<a_1<\cdots <a_n<p$ be all the cubic residues modulo p in the interval $(0,p)$ . Then clearly the sequence $a_1 \bmod p,\, a_2 \bmod p,\ldots , a_n \bmod p$ is a permutation of the sequence $g^3 \bmod p,\,g^6 \bmod p,\ldots , g^{3n} \bmod p$ . We determine the sign of this permutation.

Analysis of linear complexity of generalized cyclotomic Q-ary sequences of pn period

The article presents the analysis of the linear complexity of periodic q-ary sequences when changing k of their terms per period. Sequences are formed on the basis of new generalized cyclotomy modulo equal to the degree of an odd prime. There has been obtained a recurrence relation and an estimate of the change in the linear complexity of these sequences, where q is a primitive root modulo equal to the period of the sequence. It can be inferred from the results that the linear complexity of these sequences does not sign ificantly decrease when k is less than half the period. The study summarizes the results for the binary case obtained earlier.

PRIMES IN ARITHMETIC PROGRESSIONS AND NONPRIMITIVE ROOTS

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb{Z}/p\mathbb{Z})^{\ast },$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each coprime residue class contains a subset of primes $p$ of positive natural density which do not have $g$ as a $t$-near primitive root and we prove a more difficult variant.

The weight distributions of irreducible cyclic codes of length 2npm

Let [Formula: see text] be a primitive root modulo [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct odd primes. Let [Formula: see text] be a finite field. For such pair of [Formula: see text] and [Formula: see text], the explicit expressions of minimal and generating polynomials over [Formula: see text] are obtained for all irreducible cyclic codes of length [Formula: see text]. In Sec. 4, it is observed that the weight distributions of all irreducible cyclic codes of length [Formula: see text] over [Formula: see text] can be computed easily with the help of the results obtained in [P. Kumar, M. Sangwan and S. K. Arora, The weight distribution of some irreducible cyclic codes of length [Formula: see text] and [Formula: see text], Adv. Math. Commun. 9 (2015) 277–289]. An explicit formula is also given to compute the weight distributions of irreducible cyclic codes of length [Formula: see text] over [Formula: see text].

QUADRATIC NONRESIDUES AND NONPRIMITIVE ROOTS SATISFYING A COPRIMALITY CONDITION

Let $q\geq 1$ be any integer and let $\unicode[STIX]{x1D716}\in [\frac{1}{11},\frac{1}{2})$ be a given real number. We prove that for all primes $p$ satisfying $$\begin{eqnarray}p\equiv 1\!\!\!\!\hspace{0.6em}({\rm mod}\hspace{0.2em}q),\quad \log \log p>\frac{2\log 6.83}{1-2\unicode[STIX]{x1D716}}\quad \text{and}\quad \frac{\unicode[STIX]{x1D719}(p-1)}{p-1}\leq \frac{1}{2}-\unicode[STIX]{x1D716},\end{eqnarray}$$ there exists a quadratic nonresidue $g$ which is not a primitive root modulo $p$ such that $\text{gcd}(g,(p-1)/q)=1$.

Square-full primitive roots

We use character sum estimates to give some bounds on the least square-full primitive root modulo a prime. In particular, we show that there is a square-full primitive root mod [Formula: see text] less than [Formula: see text].