Related papers: Twelve new primitive binary trinomials

Four circulant codes form a special class of $2$-generator, index $4$, quasi-cyclic codes. Under some conditions on their generator matrices they can be shown to be self-dual. Artin primitive root conjecture shows the existence of an…

For $q$ an odd prime power with $q>169$ we prove that there are always three consecutive primitive elements in the finite field $\mathbb{F}_{q}$. Indeed, there are precisely eleven values of $q \leq 169$ for which this is false. For $4\leq…

Permutation polynomials over finite fields play important roles in finite fields theory. They also have wide applications in many areas of science and engineering such as coding theory, cryptography, combinatorial design, communication…

The order and disorder of binary representations of the natural numbers < 2^8 is measured using the BiEntropy function. Significant differences are detected between the primes and the non primes. The BiEntropic prime density is shown to be…

Permutation polynomials over finite fields constitute an active research area and have applications in many areas of science and engineering. In this paper, four classes of monomial complete permutation polynomials and one class of…

By definition primitive and $2$-primitive elements of a finite field extension $\mathbb{F}_{q^n}$ have order $q^n-1$ and $(q^n-1)/2$, respectively. We have already shown that, with minor reservations, there exists a primitive element and a…

In this paper, we present a necessary and sufficient condition for a nonnegative tensor to be a primitive one, show that the exponent set of nonnegative primitive tensors with order $m(\ge n)$ and dimension $n$ is $\{k| 1\le k\le…

Let GF(q), q=p^r, be a finite field with a primitive element g. In this paper we use exponential sums and Jacobi sums to compute the number of the irreducible polynomials of degree m over GF(q) with trace fixed and norm restricted to a…

In this paper we introduce a new class of non-amenable groups denoted by ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ which give rise to $\textit{prime}$ von Neumann algebras. This means that for every $\Gamma \in {\bf NC}_1 \cap {\bf…

We construct a primitive permutation action of the Steinberg triality group $^3D_4(2)$ of degree $4064256$ and show that there are distinct points $\alpha,\beta$ such that there is no derangement $g\in{^3D_4}(2)$ with $\alpha^g=\beta$. This…

In 1978, Allan.Wm.Johnson obtained an example of the representation of 1 by the sum of reciprocals of the product of two distinct prime numbers. His example has 48 terms [1], and we had no examples which have less than 48 terms until now.…

This paper's central theme is to prove the existence of an n-algebra whose multiplication cannot be expressed employing any binary operation. Furthermore, to prove if two algebras are not isomorphic, this property does not hold for…

Let $q\geq2$ be a prime power and consider Drinfeld modules of rank 2 over $\mathbb{F}_q[T]$. We prove that there are no points with coordinates being Drinfeld singular moduli, on a family of hyperbolas $XY=\gamma$, where $\gamma$ is a…

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p+1. An extension of…

Linearized polynomials have attracted a lot of attention because of their applications in both geometric and algebraic areas. Let $q$ be a prime power, $n$ be a positive integer and $\sigma$ be a generator of…

This paper is a complement to our previous paper [21]. It surveys the works on the Furstenberg set $S=\{2^{m}3^{n}: n\ge 0, m\ge 0\}$ and its random version $T$. We also present some new results. For example, it is proved that $T$ almost…

In 2000 Deaconescu raised a question whether there exists a composite $n$ for which $S_2(n)|\phi(n)-1$, where $\phi(n)$ is Euler's function and $S_2(n)$ is Schemmel's totient function. In this paper we prove that any such $n$ is odd,…

We prove the infinitude of shifted primes $p-1$ without prime factors above $p^{0.2844}$. This refines $p^{0.2961}$ from Baker and Harman in 1998. Consequently, we obtain an improved lower bound on the the distribution of Carmichael…

We survey the classical results on the prime number theorem

In this paper we investigate some new problems in additive combinatorics. Our problems mainly involve permutations (or circular permutations) $n$ distinct numbers (or elements of an additive abelian group) $a_1,\ldots,a_n$ with adjacent…