Related papers: Conjectures about Primes and Cyclic Numbers

Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. The first two authors showed in part I that if m is a prime power and n and k range over the non-negative integers, then a(mn,k) assumes every integer value. Here this…

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n \geq 1$, the interval $(n^2,(n+1)^2)$ contains an integer having at most $3$ prime factors, counted with multiplicity. This improves the…

A group G is almost cyclic if there is an element x in G, such that for all g in G, there is an element y in G and an integer n with ygy^{-1} = x^n (that is, every element is conjugate to some power of x). W. Ziller asked whether there are…

We present a survey of exact and asymptotic formulas on the number of cyclic subgroups and total number of subgroups of the groups ${\Bbb Z}_{n_1} \times \cdots \times {\Bbb Z}_{n_k}$, where $k\ge 2$ and $n_1,\ldots,n_k$ are arbitrary…

We introduce and study the concept of cyclicity degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be cyclic. Explicit formulas are obtained for some particular classes of finite groups. An…

A partition of a positive integer $n$ is a representation of $n$ as a sum of a finite number of positive integers (called parts). A trapezoidal number is a positive integer that has a partition whose parts are a decreasing sequence of…

Quite recently, in [8] the authoor of this paper considered the distribution of primes in the sequence $(S_n)$ whose $n$th term is defined as $S_n=\sum_{k=1}^{2n}p_k$, where $p_k$ is the $k$th prime. Some heuristic arguments and the…

In this paper we study $\prod\limits_{i=1}^{n} \mathbb{Z}_{2^i}$-Additive Cyclic Codes. These codes are identified as $\mathbb{Z}_{2^n}[x]$-submodules of $\prod\limits_{i=1}^{n}\mathbb{Z}_{2^i}[x]/ \langle x^{\alpha_i}-1\rangle$; $\alpha_i$…

We consider the representation of primes as a sum of a prime and twice a triangular number. We prove that a subset of the primes having density 1 is expressible in this form. We conjecture that every odd prime number is expressible as a sum…

We prove that every sufficiently large integer $n$ can be written as the sum of a prime and an integer that is not square-free. In addition, we expect this result holds for every $n > 24$ and prove two results to support this claim. First,…

New Mersenne conjectures. The problems of simplicity, common prime divisors and free from squares of numbers $L(n) = 2^{2n}\pm2^n\pm1$ are investigated. Wonderful formulas $gcd $ for numbers $L (n) $ and numbers repunit are proved.

We prove that an inseparable graph can have any positive number of cycles with the six exceptions 2, 4, 5, 8, 9, 16, and that an inseparable cubic graph has the additional exceptions 1 and 13. The exceptions for simple inseparable cubic…

This is a survey article on prime number races. Chebyshev noticed in the first half of the nineteenth century that for any given value of x, there always seem to be more primes of the form 4n+3 less than x then there are of the form 4n+1.…

One of the classical problems in group theory is determining the set of positive integers $n$ such that every group of order $n$ has a particular property $P$, such as cyclic or abelian. We first present the Sylow theorems and the idea of…

It is well-known that for any distinct positive integers $k$ and $n$, the numbers $2^{2^k}+1$ and $2^{2^n}+1$ are relatively prime. In this paper we consider the situation when 1 is replaced by some positive integer $d>1$

Let $A$ be a set of natural numbers and let $S_{n,A}$ be the set of all permutations of $[n]=\{1,2,...,n\}$ with cycle lengths belonging to $A$. Furthermore, let $\mid A(n)\mid$ denote the cardinality of the set $A(n)=A\cap [n]$. The limit…

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Laba and Meyerowitz, but the spectral-tile direction is far from being well-understood. The product of at most…

In this paper convolutional codes with cyclic structure will be investigated. These codes can be understood as left principal ideals in a suitable skew-polynomial ring. It has been shown in [3] that only certain combinations of the…

Prime number multiplet classifications and patterns are extended to negative integers. The extension from prime numbers to single prime powers is also studied. Prime number septets at equal distance are given. It is also shown that each…

In 1996, in his last paper, Erd\H{o}s asked the following question that he formulated together with Faudree: is there a positive $c$ such that any $(n+1)$-regular graph $G$ on $2n$ vertices contains at least $c 2^{2n}$ distinct…