Related papers: Construction of normal numbers via generalized pri…

In this work we present a natural generalization of strong pseudoprime to base $b$, which we have called $q$-pseudoprime to base $b$. It allows us to present another way to define a Midy's number to base $b$ (overpseudoprime to base $b$).…

Given an integer $b\geqslant 2$ and a set $P$ of prime numbers, the set $T_P $ of Toeplitz numbers comprises all elements of $[0,b[$ whose digits $(a_n)_{n\geqslant 1}$ in the base-$b$ expansion satisfy $a_n=a_{pn}$ for all $p\in P$ and…

An element $\alpha \in \mathbb {F}_{q^n}$ is normal over $\mathbb {F}_q$ if $\alpha$ and its conjugates $\alpha, \alpha^q, \cdots \alpha^{q^{n-1}}$ form a basis of $\mathbb {F}_{q^n}$ over $\mathbb {F}_q$. Recently, Huczynska, Mullen,…

We give the q-analogue of the sums of the n-th powers of positive integers up to k-1.

In this note, we approximate the average of prime powers in the decomposition of $n!$ into prime numbers.

For $\alpha>1$ we represent a real number in $(0,1]$ in the form \[ \sum_{i=1}^{\infty}(\alpha-1)^{i-1}\alpha^{-(d_{1}+\dots+d_{i})}\] with $d_{i}\in\mathbb{N}$. We discuss ergodic theoretical and dimension theoretical aspects of this…

In this paper some generalizations of the sum of powers of natural numbers is considered. In particular, the class of sums whose generating function is the power of the generating function for the classical sums of powers is studying. The…

A Generalized Numeration Base is defined in this paper, and then particular cases are presented, such as Prime Base, Square Base, m-Power Base, Factorial Base, and operations in these bases. These bases are important for partitions of…

We consider the question of approximating any real number $\alpha$ by sums of $n$ rational numbers $\frac{a_1}{q_1} + \frac{a_2}{q_2} + ... + \frac{a_n}{q_n}$ with denominators $1 \leq q_1, q_2, ..., q_n \leq N$. This leads to an inquiry on…

The purpose of this paper is to define generalized twisted q-Bernoulli numbers by using p-adic q-integrals. Furthermore, we construct a q-analogue of the p-adic generalized twisted L-functions which interpolate generalized twisted…

Let $\mathbb{Q}(\alpha)$ and $\mathbb{Q}(\beta)$ be linearly disjoint number fields and let $\mathbb{Q}(\theta)$ be their compositum. We prove that the first-degree prime ideals of $\mathbb{Z}[\theta]$ may almost always be constructed in…

Adler, Keane, and Smorodinsky showed that if one concatenates the finite continued fraction expansions of the sequence of rationals \[ \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{2}{4}, \frac{3}{4}, \frac{1}{5}, \cdots \] into…

We prove that digital sequences modulo $m$ along squares are normal, which covers some prominent sequences like the sum of digits in base $q$ modulo $m$, the Rudin-Shapiro sequence and some generalizations. This gives, for any base, a class…

We study three different $q$-analogues of the harmonic numbers. As applications, we present some generating functions involving number theoretical functions and give the $q$-generalization of Gosper's exponential generating function of…

We prove that the set of normalized differences between primes, defined as $S = \{(p-q)/(p+q) : p > q \text{ are primes}\}$, is dense in the open unit interval $(0,1)$. Our proof provides an explicit construction algorithm with quantitative…

The $R_{\sigma,t}$-transform introduced by Bassa and Menares can be used to construct families of irreducible polynomials in $\mathbb{F}_q[x]$. This iterative construction is a generalization of Cohen's $R$-transform. For this transform,…

In the paper, from the point of view of recurrent numbers of the Jacobsthal type, the Collatz problem with the general aq+-1 function of conjecture odd positive integers q from the set of natural numbers is investigated. Formulated…

Let $q$ be a prime power. We construct stable polynomials of the form $b^{m-1}(x+a)^m+c(x+a)+d$ over a finite field $\mathbb{F}_{q}$ for $m=2,3,4$ by Capelli's lemma. When $m=3$ and $q$ is even, we confirm the conjecture of Ahmadi and…

For $q=3^r$ ($r>0$), denote by $\mathbb{F}_q$ the finite field of order $q$ and for a positive integer $m\geq2$, let $\mathbb{F}_{q^m}$ be its extension field of degree $m$. We establish a sufficient condition for existence of a primitive…

We show that an iterative method for computing the center of mass (CM) of q units of mass, placed on a unit interval [0, 1] along the x- axis, give rise to a simple procedure for expanding rational numbers less than unity in powers of r/s <…