Related papers: Additive uniqueness of $\mathtt{PRIMES}-1$ for mul…

Let $F=\mathbb{F}_{q^m}$, $m>6$, $n$ a positive integer, and $f=p/q$ with $p$, $q$ co-prime irreducible polynomials in $F[x]$ and deg$(p)$ $+$ deg$(q)= n$. A sufficient condition has been obtained for the existence of primitive pairs…

For a fixed polynomial $\Delta$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+\Delta$ are both irreducible, an $\mathbb F_q[T]$-analogue of the twin primes problem. In the large-$q$ limit, we…

Let $\mathcal{P}$ be the set of the primes. We consider a class of random multiplicative functions $f$ supported on the squarefree integers, such that $\{f(p)\}_{p\in\mathcal{P}}$ form a sequence of $\pm1$ valued independent random…

Let $\mathbb F_q$ be the finite field with $q$ elements, where $q$ is a prime power and, for each integer $n\ge 1$, let $\mathbb F_{q^n}$ be the unique $n$-degree extension of $\mathbb F_q$. The $\mathbb F_q$-orders of an element in…

In the present paper we prove that there exist infinitely many arithmetic progressions of three different primes $p_1,p_2,p_3=2p_2-p_1$ such that $p_1=x_1^2 + y_1^2 +1$, $p_2=x_2^2 + y_2^2 +1$.

A parking function of length $n$ is prime if we obtain a parking function of length $n-1$ by deleting one 1 from it. In this note we give a new direct proof that the number of prime parking functions of length $n$ is $(n-1)^{n-1}$. This…

Let $a$ and $b$ be positive integers with prime factorisations $a = p_1^np_2^n$ and $b = q_1^nq_2^n$. We prove that the number of essentially distinct $\alpha$-graceful labelings of the complete bipartite graph $K_{a, b}$ equals the…

After obtaining some useful identities, we prove an additional functional relation for $q$ exponentials with reversed order of multiplication, as well as the well known direct one in a completely rigorous manner.

Let $A$ be a centrally closed prime algebra over a characteristic 0 field $k$, and let $q:A\to A$ be the trace of a $d$-linear map (i.e., $q(x)=M(x,...,x)$ where $M:A^d\to A$ is a $d$-linear map). If $[q(x),x]=0$ for every $x\in A$, then…

Power functions with low $c$-differential uniformity have been widely studied not only because of their strong resistance to multiplicative differential attacks, but also low implementation cost in hardware. Furthermore, the…

Let $\Bbb F_q$ be the finite field with $q$ elements and let $p=\text{char}\,\Bbb F_q$. It was conjectured that for integers $e\ge 2$ and $1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots+X^{q^a-2}$ is a permutation polynomial of…

We determine all entire functions $f$ such that for nonzero complex values $a\neq b$ the implications $f=a \Rightarrow f' =a$ and $f' =b \Rightarrow f=b$ hold. This solves an open problem in uniqueness theory. In this context we give a…

In this note we determine all power series $F(X)\in 1+X\F_p[[X]]$ such that $(F(X+Y))^{-1} F(X)F(Y)$ has only terms of total degree a multiple of $p$. Up to a scalar factor, they are all the series of the form $F(X)=E_p(cX)\cdot G(X^p)$ for…

In this paper, we study the unicity of entire functions concerning their $q-$shifts and $k-$th derivatives and prove: Let $f(z)$ be a transcendental entire function of zero-order, and $g(z)$ define as in (1.1). Let $a(z), b(z)$ be two…

Let $M_n(\mathbb{F})$ be the algebra of $n \times n$ matrices over a field $\mathbb{F}$ of characteristic not equal to $2$. If $n\ge 2$, we show that an arbitrary map $\phi : M_n(\mathbb{F}) \to M_n(\mathbb{F})$ is Jordan multiplicative,…

Let $G = {\rm U}(2m, {\mathbb F}_{q^2})$ be the finite unitary group, with $q$ the power of an odd prime $p$. We prove that the number of irreducible complex characters of $G$ with degree not divisible by $p$ and with Frobenius-Schur…

In this paper we investigate continuity properties of functions $f:\mathbb{R}_+\to\mathbb{R}_+$ that satisfy the $(p,q)$-Jensen convexity inequality $$ f\big(H_p(x,y)\big)\leq H_q(f(x),f(y)) \qquad(x,y>0), $$ where $H_p$ stands for the…

We show that for any polynomial $f: \mathbb{Z}\to \mathbb{Z}$ with positive leading coefficient and irreducible over $\mathbb{Q}$, if $N$ is large enough then there are two strings of consecutive positive integers $I_{1}=\{n_1-m,\ldots,…

The subset of quadratic primes {p = an^2 + bn + c : n => 1} generated by an irreducible polynomial f(x) = ax^2 + bx + c over the integers is widely believed to be an unbounded subset of prime numbers. This note provides the details of a…

Let $q$ be a prime power, $2\le r\le q$, and $f=a{\tt X}+{\tt X}^{r(q-1)+1}\in\Bbb F_{q^2}[{\tt X}]$, where $a\ne 0$. The conditions on $r,q,a$ that are necessary and sufficient for $f$ to be a permutation polynomial (PP) of ${\Bbb…