Related papers: Generalized Mersenne Numbers of the form $cx^2$

The set of prime numbers has been analyzed, based on their algebraic and arithmetical structure. Here by obtaining a sort of linear formula for the set of prime numbers, they are redefined and identified; under a systematic procedure it has…

The goal of this note is to provide an alternative proof of Theorem 1.1 (i) in [4], that is, if $n\geq 2$ and $M^{\alpha}$ is bounded on $L^{p}(\mathbb{R}^{n})$ for some $\alpha\in \mathbb{C}$ and $p\geq 2$, then we have \begin{align*}…

The general position number ${\rm gp}(G)$ of a connected graph $G$ is the cardinality of a largest set $S$ of vertices such that no three distinct vertices from $S$ lie on a common geodesic; such sets are refereed to as gp-sets of $G$. The…

We propose a generalization of Carmichael numbers, where the multiplicative group $\mathbb G_\mathrm{m} = \mathrm{GL}(1)$ is replaced by $\mathrm{GL}(m)$ for $m\geq 2$. We prove basic properties of these families of numbers and give some…

For each natural number $m\ge 3$, let $P_m(x)$ denote the generalized $m$-gonal number $\frac{(m-2)x^2-(m-4)x}{2}$ with $x\in\mathbb{Z}$. In this paper, with the help of the congruence theta function, we establish conditions on $a$, $b$,…

Let $s(m,n)$ denote the classical \DED sum, where $n$ is a positive integer and $m\in\{0,1,\ldots, n-1\}$, $(m,n)=1$. For a given positive integer $k$, we describe a set of at most $k^2$ numbers $m$ for which $s(m,n)$ may be $\ge s(k,n)$,…

In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…

This article determines a lower bound for the number Germain primes $p$ and $2p+1$ up to a large number $x$.

For a normed linear space $(X,|\cdot|)$ and $p>0$ we characterize all $n$-tuples $(\mu_1,...,\mu_n)\in\mathbb{R}^{n}$ for which the generalized triangle inequality of the second type…

Let $c(x_1,...,x_d)$ be a multihomogeneous central polynomial for the $n\times n$ matrix algebra $M_n(K)$ over an infinite field $K$ of positive characteristic $p$. We show that there exists a multihomogeneous polynomial $c_0(x_1,...,x_d)$…

We consider the integers having the property of reversing when multiplied by a specific integer k. First, we proved that k should be either 1, 4 or 9. Second, we classify these integers as (10, 1)- reverse multiples, (10, 4)- reverse…

All integer solutions $\left(M,a,c\right)$ to the problem of the sums of $M$ consecutive cubed integers $\left(a+i\right)^{3}$ ($a>1$, $0\leq i\leq M-1$) equaling squared integers $c^{2}$ are found by decomposing the product of the…

Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer $a$, there are infinitely many $n\in \mathbb N$ such that for each prime factor $p|n$, we have $p-a|n-a$. This can be…

Let $p \geqslant 3$ be a prime number and let $n \geqslant 0$ be an integer such that $p-1$ divides $n.$ In this short note we construct a family of $(p,n)$-gonal Riemann surfaces of maximal genus $2np+(p-1)^2$ with more than one…

Due to their broad application to different fields of theory and practice, generalized Petersen graphs $GPG(n,s)$ have been extensively investigated. Despite the regularity of generalized Petersen graphs, determining an exact formula for…

We define Poisson genericity for infinite sequences in any finite or countable alphabet with an invariant exponentially-mixing probability measure. A sequence is Poisson generic if the number of occurrences of blocks of symbols…

For a positive integer $m>1$, if the generalized Markoff equation $a^2+b^2+c^2=3abc+m$ has a solution triple, then it has infinitely many solutions. We show that all positive solution triples are generated by a finite set of triples that we…

Mersenne primes and Fermat primes may be thought of as primes of the form $\Phi_m(2)$, where $\Phi_m(x)$ is the $m$th cyclotomic polynomial. This paper discusses the more general problem of primes and composites of this form.

Let $m\ge3$ be an integer. The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\binom n2+n$ $(n=0,1,2,\ldots)$. A famous claim of Fermat proved by Cauchy asserts that each nonnegative integer is the sum of $m+2$ polygonal numbers…

In this work, the author shows a sufficient and necessary condition for an integer of the form $(zn-y^n)/(z-y)$ to be divisible by some perfect $mth$ power $p$, where $p$ is an odd prime and $m$ is a positive integer. A constructive method…