Related papers: The exceptional set in a generalized Goldbach`s pr…

In this paper, we strengthen a result by Green about an analogue of Sarkozy's theorem in the setting of polynomial rings $\mathbb{F}_q[x]$. In the integer setting, for a given polynomial $F \in \mathbb{Z}[x]$ with constant term zero, (a…

For a fixed quadratic irreducible polynomial $f$ with no fixed prime factors at prime arguments, we prove that there exist infinitely many primes $p$ such that $f(p)$ has at most 4 prime factors, improving a classical result of Richert who…

A (left) Engel sink of an element g of a group G is a subset containing all sufficiently long commutators [...[[x,g],g],...,g], where x ranges over G. We prove that if p is a prime and G a finite group in which, for some positive integer m,…

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error…

Given a compact subset $\Sigma$ of the real numbers obeying some technical conditions, we consider the set of algebraic integers whose conjugates all lie in $\Sigma$. The distribution of conjugates of such an integer defines a probability…

Let G be a finitely presented group, and let p be a prime. Then G is 'large' (respectively, 'p-large') if some normal subgroup with finite index (respectively, index a power of p) admits a non-abelian free quotient. This paper provides a…

We consider a generic family of polynomial maps $f:=(f_1,f_2):\mathbb{C}^2\rightarrow\mathbb{C}^2$ with given supports of polynomials, and degree $ d(f):=\max (deg f_1, deg f_2)$. We show that the (non-) properness of maps $f$ in this…

For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of…

Let $G$ denote a compact monothetic group, and let $$\rho (x) = \alpha_k x^k + \ldots + \alpha_1 x + \alpha_0,$$ where $\alpha_0, \ldots , \alpha_k$ are elements of $G$ one of which is a generator of $G$. Let $(p_n)_{n\geq 1}$ denote the…

For $m$ an even positive integer and $p$ a prime, we show that the generalized Euler polynomial $E_{mp}^{(mp)}(x)$ is in Eisenstein form with respect to $p$ if and only if $p$ does not divide $m (2^m-1)B_m$. As a consequence, we deduce that…

In this paper we present some observations about the well-known Goldbach conjecture. In particular we list and interpret some numerical results which allow us to formulate a relation between prime numbers and even integers. We can also…

We obtain various irreducibility criteria for pairs of polynomials $(f(X),g(X))$ with integer coefficients whose resultant $Res(f,g)$ is a prime number, or is divisible by a sufficiently large prime number, and also for some of their linear…

Let $f$ be a primitive positive definite integral binary quadratic form of discriminant $-D$ and let $\pi_f(x)$ be the number of primes up to $x$ which are represented by $f$. We prove several types of upper bounds for $\pi_f(x)$ within a…

In this paper, we give decision criteria for normal binomial difference polynomial ideals in the univariate difference polynomial ring F{y} to have finite difference Groebner bases and an algorithm to compute the finite difference Groebner…

Let $p$ be a prime number, and $h$ a positive integer such that $\gcd(p,h)=1$. We prove, without invoking Dirichlet's theorem, that the arithmetic progression $p\left(\mathbf{N}\cup \{0\}\right)+h$ contains infinitely many prime numbers.…

Suppose L is any finite algebraic extension of either the ordinary rational numbers or the p-adic rational numbers. Also let g_1,...,g_k be polynomials in n variables, with coefficients in L, such that the total number of monomial terms…

Let A be a finite set of integers. For a polynomial f(x_1,...,x_n) with integer coefficients, let f(A) = {f(a_1,...,a_n) : a_1,...,a_n \in A}. In this paper it is proved that for every pair of normalized binary linear forms f(x,y)=u_1x+v_1y…

Due to the distribution of primes among integers, we establish an upper bound for the probability $\mathbb{P}_n$ that the Goldbach conjecture fails. Assuming the conjecture holds true for all even number less than $2N$, we prove this…

In this paper we continue our study, begun in part I, of the exceptional set of integers, not restricted by elementary congruence conditions, which cannot be represented as sums of three or four squares of primes. We correct a serious…

Let q be an odd power of a prime p and let A/Fq be a supersingular abelian variety of dimension g. We show that if p>2g+1, then the characteristic polynomial of the q-Frobenius is an even polynomial. This generalizes the well-known result…