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 $<a> \subseteq \mathbb F_p^*.$

An Efficient Approach to Point-Counting on Elliptic Curves from a Prominent Family over the Prime Field Fp

Here, we elaborate an approach for determining the number of points on elliptic curves from the family Ep={Ea:y2=x3+a(modp),a≠0}, where p is a prime number >3. The essence of this approach consists in combining the well-known Hasse bound with an explicit formula for the quantities of interest-reduced modulo p. It allows to advance an efficient technique to compute the six cardinalities associated with the family Ep, for p≡1(mod3), whose complexity is O˜(log2p), thus improving the best-known algorithmic solution with almost an order of magnitude.

Generator of PRN's on the norm group

Let $p$ be a prime number, $d\in\mathds{N}$, $\left(\frac{-d}{p}\right)=-1$, $m>2$, and let $E_m$ denotes the set of of residue classes modulo $p^m$ over the ring of Gaussian integers in imaginary quadratic field $\mathds{Q}(\sqrt{-d})$ with norms which are congruented with 1 modulo $p^m$. In present paper we establish the polynomial representations for real and imagimary parts of the powers of generating element $u+iv\sqrt{d}$ of the cyclic group $E_m$. These representations permit to deduce the ``rooted bounds'' for the exponential sum in Turan-Erd\"{o}s-Koksma inequality. The new family of the sequences of pseudo-random numbers that passes the serial test on pseudorandomness was being buit.

The Legendre’s Symbol Modulo p and Its Some Elementary Results

The main purpose of this article is using the elementary methods and the properties of the character sums to study the calculating problem of the number of the solutions for one kind congruence equation modulo p (an odd prime) and give some interesting identities and asymptotic formulas for it.

Construction and classification of \(p\)-ring class fields modulo \(p\)-admissible conductors

Each \(p\)-ring class field \(K_f\) modulo a \(p\)-admissible conductor \(f\) over a quadratic base field \(K\) with \(p\)-ring class rank \(\varrho_f\) mod \(f\) is classified according to Galois cohomology and differential principal factorization type of all members of its associated heterogeneous multiplet \(\mathbf{M}(K_f)=\lbrack(N_{c,i})_{1\le i\le m(c)}\rbrack_{c\mid f}\) of dihedral fields \(N_{c,i}\) with various conductors \(c\mid f\) having \(p\)-multiplicities \(m(c)\) over \(K\) such that \(\sum_{c\mid f}\,m(c)=\frac{p^{\varrho_f}-1}{p-1}\). The advanced viewpoint of classifying the entire collection \(\mathbf{M}(K_f)\), instead of its individual members separately, admits considerably deeper insight into the class field theoretic structure of ring class fields. The actual construction of the multiplet \(\mathbf{M}(K_f)\) is enabled by exploiting the routines for abelian extensions in the computational algebra system Magma.