Related papers: Primitive sets and an Euler phi function for subse…

In this article, we study the Euler's factorial series $F_p(t)=\sum_{n=0}^\infty n!t^n$ in $p$-adic domain under the Generalized Riemann Hypothesis. First, we show that if we consider primes in $k\varphi(m)/(k+1)$ residue classes in the…

Let $\mathcal A$ be a star-shaped polygon in the plane, with rational vertices, containing the origin. The number of primitive lattice points in the dilate $t\mathcal A$ is asymptotically $\frac6{\pi^2}$ Area$(t\mathcal A)$ as $t\to…

We introduce a sequence representation of a random variable $X$ supported on a compact interval $[a,b]$, which we call a primitive sequence. We construct this sequence by repeatedly antidifferentiating the associated cumulative distribution…

Let $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second…

A transitive permutation group is said to be semiprimitive if each of its normal subgroups is either semiregular or transitive.The class of semiprimitive groups properly contains primitive groups, quasiprimitive groups and innately…

For a class of Lucas sequences ${x_n}$, we show that if $n$ is a positive integer then $x_n$ has a primitive prime factor which divides $x_n$ to an odd power, except perhaps when $n = 1, 2, 3 or 6$. This has several desirable consequences.

Fix $A$, a family of subsets of natural numbers, and let $G_A(n)$ be the maximum cardinality of a subset of $\{1,2,..., n\}$ that does not have any subset in $A$. We consider the general problem of giving upper bounds on $G_A(n)$ and give…

Let $n$ be an arbitrary integer, let $p$ be a prime factor of $n$. Denote by $\omega_1$ the $p^{th}$ primitive unity root, $\omega_1:=e^{\frac{2\pi i}{p}}$. Define $\omega_i:=\omega_1^i$ for $0\leq i\leq p-1$ and…

For a natural number $k>1$, let $f_k(n)$ denote the number of distinct representations of a natural number $n$ of the form $p^k+q^k$ for primes $p,q$. We prove that, for all $k>1$, $$\limsup_{n\to\infty}f_k(n)=\infty.$$ This positively…

Let $\phi(n)$denote Euler's phi function. We study the distribution of the numbers $gcd(n,\phi(n))$ and their divisors. Our results generalize previous results of Erd\H{o}s and Pollack.

For $x\ge0$ let $\pi(x)$ be the number of primes not exceeding $x$. The asymptotic behaviors of the prime-counting function $\pi(x)$ and the $n$-th prime $p_n$ have been studied intensively in analytic number theory. Surprisingly, we find…

Denote by $\tau$ k (n), $\omega$(n) and $\mu$ 2 (n) the number of representations of n as product of k natural numbers, the number of distinct prime factors of n and the characteristic function of the square-free integers, respectively. Let…

This paper presents a primitive $n$th root of unity in $\mathbb C$. The approach is very elementary and avoids the following: the complex exponential function, trigonometry, and group theory. It also avoids differentiation, integration, and…

Given an explicit presentation of a reflection group of rank two (or any rank two group for that matter), we give a simple procedure for calculating all its systems of imprimitivity, when viewed as a matrix group over the quaternions. This…

In this paper, we consider rational functions $f$ with some minor restrictions over the finite field $\mathbb{F}_{q^n},$ where $q=p^k$ for some prime $p$ and positive integer $k$. We establish a sufficient condition for the existence of a…

A primitive root modulo an integer $n$ is the generator of the multiplicative group of integers modulo $n$. Gauss proved that for any prime number $p$ greater than $3$, the sum of its primitive roots is congruent to $1$ modulo $p$ while its…

Working in a theory with an integer-valued dimension on interpretable sets, we classify pseudofinite definably primitive permutation groups acting on one-dimensional sets which satisfy a version of chain condition on centralizers and on…

Let $q=p^k$ be a prime power, let $n\geq2$ be an integer and let $\mathbb{F}_{q^n}$ be a finite field. It is shown that the set of primitive normal elements is a Salem set. Furthermore, it is proved that this set is strongly equidistributed…

Four functions counting the number of subsets of $\{1, 2, ..., n\}$ having particular properties are defined by Nathanson and generalized by many authors. They derive explicit formulas for all four functions. In this paper, we point out…

We prove that every group ring of a non-abelian locally free group which is the union of an ascending sequence of free groups is primitive. In particular, every group ring of a countable non-abelian locally free group is primitive. In…