Related papers: One-Step $G$-Unimprovable Numbers

Let $G$ be a finite abelian group. The critical number ${\rm cr}(G)$ of $G$ is the least positive integer $\ell$ such that every subset $A\subseteq G\setminus\{0\}$ of cardinality at least $\ell$ spans $G$, i.e., every element of $G$ can be…

Let $p_1=2, p_2=3, p_3=5, \ldots$ be the consecutive prime numbers, $S_n$ the numerical semigroup generated by the primes not less than $p_n$ and $u_n$ the largest irredundant generator of $S_n$. We will show, that $\bullet$ $u_n\sim3p_n$.…

Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)\leq \frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as…

Let $G(n)=\sigma (n)/(n \log \log n )$. Robin made hypothesis that $G(n)<e^\gamma$ for all integer $n>5040$. If there exists counterexample to Robin hypothesis, then there must exist finite number of counterexamples $n>5040$ such that…

We study finite groups $G$ with the property that for any subgroup $M$ maximal in $G$ whose order is divisible by all the prime divisors of $|G|$, $M$ is supersolvable. We show that any nonabelian simple group can occur as a composition…

Let $\{q_n\}_{n=0}^\infty\subset [0,1]$ satisfy $q_0=0$, $\sum_{n=0}^\infty q_n=1$, and $\gcd\{n\geq 1\mid q_n\neq 0\}=1$. We consider the following process: Let $x$ be a real number. We first set $x=0$. Then $x$ is increased by $i$ with…

A cyclic subgroup $N$ of a finite group $G$ is called a uni-width subgroup of $G$ if $N$ is the unique cyclic subgroup of $G$ of order $|N|$. In this article, we prove that a finite group $G$ admits a unique largest uni-width subgroup…

Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbers by $G_n:=\ell(1-\ell^n)B_n$, with $B_n$ the $n$-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes.…

Denote the sum of element orders in a finite group $G$ by $\psi(G)$ and let $C_n$ denote the cyclic group of order $n$. In this paper, we prove that if $|G|=n$ and $\psi(G)>\frac{13}{21}\,\psi(C_n)$, then $G$ is nilpotent. Moreover, we have…

For a real set $A$ consider the semigroup $S(A)$, additively generated by $A$; that is, the set of all real numbers representable as a (finite) sum of elements of $A$. If $A \subset (0,1)$ is open and non-empty, then $S(A)$ is easily seen…

Let $k$ be an algebraically closed field of characteristic $p$, possibly zero, and $G=q$-$\GL_3(k)$, the quantum group of three by three matrices as defined by Dipper and Donkin. We may also take $G$ to be $\GL_3(k)$. We first determine the…

We prove that there are infinitely many integers $n$ such that $n$ and $n+1$ have the same number of distinct prime divisors.

Let $q$ be a power of a prime $p$ and let $U(q)$ be a Sylow $p$-subgroup of a finite Chevalley group $G(q)$ defined over the field with $q$ elements. We first give a parametrization of the set $\text{Irr}(U(q))$ of irreducible characters of…

The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists and…

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…

Suppose $G$ is a real reductive group. The determination of the irreducible unitary representations of $G$ is one of the major unsolved problem in representation theory. There is evidence to suggest that every irreducible unitary…

Suppose that $n$ is $0$ or $4$ modulo $6$. We show that there are infinitely many primes of the form $p^2 + nq^2$ with both $p$ and $q$ prime, and obtain an asymptotic for their number. In particular, when $n = 4$ we verify the `Gaussian…

We obtain upper bounds on the number of irreducible and extended irreducible Goppa codes over $GF(p)$ of length $q$ and $q+1$, respectively defined by polynomials of degree $r$, where $q=p^t$ and $r\geq 3$ is a positive integer.

Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length…

Iannucci considered the positive divisors of a natural number $n$ that do not exceed $\sqrt{n}$ and found all forms of numbers whose such divisors are in arithmetic progression. In this paper, we generalize Iannucci's result by excluding…