Related papers: Four consecutive primitive elements in a finite fi…

It is proved that for any two subsets $A$ and $B$ of an arbitrary finite field $\Fq$ such that $|A||B|>q$ the identity $16AB=\Fq$ holds. Moreover, it is established that for every subsets $X, Y\subset \Fq$ with the property $|X||Y|\geqslant…

For any positive integers $q$, $n$, $m$ with $q$ being a prime power and $n \geq 5$, we establish a condition sufficient to ensure the existence of a primitive normal pair $(\epsilon,f(\epsilon))$ in $\mathbb{F}_{q^{n}}$ over…

It has been a well-known fact since Euclid's time that there exist infinitely many rational primes. Two natural questions arise: In which other rings, sufficiently similar to the integers, are there infinitely many irreducible elements? Is…

Let $q, n, m \in \mathbb{N}$ be such that $q$ is a prime power and $a, b \in \mathbb{F}$. In this article we establish a sufficient condition for the existence of a primitive normal pair $(\alpha, f(\alpha)) \in \mathbb{F}_{q^m}$ over…

We prove the following function field analog of the Hardy-Littlewood conjecture (which generalizes the twin prime conjecture) over large finite fields. Let n,r be positive integers and q an odd prime power. For distinct polynomials a_1,…

For n=1,2,3,... let N_n(q) denote the number of monic irreducible polynomials over the finite field F_q. We mainly show that the sequence N_n(q)^{1/n} (n>e^{3+7/(q-1)^2}) is strictly increasing and the sequence…

The results for the fractional sequence $\left \{[x/n]+1:n \leq x\right \}$, and the fractional sequence in arithmetic progression $\left \{q[x/n]+a:n \leq x\right \}$, where $a<q$ are integers such that $\gcd(a,q)=1$, prove that these…

We prove that for every field k and every positive integer n, there exists an absolutely simple n-dimensional abelian variety over k. We also prove an asymptotic result for finite fields: For every finite field k and positive integer n, we…

A (positive definite and integral) quadratic form $f$ is said to be $\textit{universal}$ if it represents all positive integers, and is said to be $\textit{primitively universal}$ if it represents all positive integers primitively. We also…

Let $\mathbb{F}_q$ stand for the finite field of odd characteristic $p$ with $q$ elements ($q=p^{n},n\in \mathbb{N} $) and $\mathbb{F}_q^*$ denote the set of all the nonzero elements of $\mathbb{F}_{q}$. Let $m$ and $t$ be positive…

Let $q$ be a prime power and $F_4(q)$ be the Chevalley group of type $F_4$ over a finite field with $q$ elements. Marcelo--Shinoda (1995) determined the values of the unipotent characters of $F_4(q)$ on all unipotent elements, extending…

For a positive integer $m$, a (positive definite integral) quadratic form is called primitively $m$-universal if it primitively represents all quadratic forms of rank $m$. It was proved in arXiv:2202.13573 that there are exactly $107$…

We give the first general constructions of even length quaternary Legendre pairs: there is a quaternary Legendre pair of length $(q-1)/2$ for every prime power $q$ congruent to $1$ modulo $4$, and there is a quaternary Legendre pair of…

We show that there are only finitely many finite fields whose members are the sum of an $n$-potent element and a $5$-potent element. Combining this with the algorithmic results provided by S.D. Cohen {\it et al.}, we confirm in the…

Let $\mathbb{F}_q$ denote the finite field of order $q,$ $n$ be a positive integer coprime to $q$ and $t \geq 2$ be an integer. In this paper, we enumerate all the complementary-dual cyclic $\mathbb{F}_q$-linear $\mathbb{F}_{q^t}$-codes of…

Let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a power of a prime. We discuss recursive methods for constructing irreducible polynomials over $\mathbb F_q$ of high degree using rational transformations. In particular,…

For positive integers $q$, Dirichlet's theorem states that there are infinitely many primes in each reduced residue class modulo $q$. A stronger form of the theorem states that the primes are equidistributed among the $\varphi(q)$ reduced…

Let $F_{q}$ be any finite field of characteristic $p>0$ having $q = p^{n}$ elements. In this paper, we have obtained the complete structure of unit groups of group algebras $F_{q}[QD_{2^k}]$, for $k = 4$ and $5$, for any prime $p>0$, where…

In this paper we establish function field versions of two classical conjectures on prime numbers. The first says that the number of primes in intervals (x,x+x^epsilon] is about x^epsilon/log x and the second says that the number of primes…

Let $\alpha:\mathbb{F}_q\to\mathbb{F}_q$ be a permutation and $\Psi(\alpha)$ be the number of collinear triples in the graph of $\alpha$, where $\mathbb{F}_q$ denotes a finite field of $q$ elements. When $q$ is odd Cooper and Solymosi once…