Related papers: Multiplicative decomposition of arithmetic progres…

In a paper published by this author in www.academia.edu(see reference[3]), it was established that there exist no three positive integers which are consecutive terms of an arithmetic progression; and whose sum of squares is a perfect or…

Let k be a number field, and denote by k^[d] the compositum of all degree d extensions of k in a fixed algebraic closure. We first consider the question of whether all algebraic extensions of k of degree less than d lie in k^[d]. We show…

Let A \subseteq [1,..,N]^2 be a set of cardinality at least N^2/(log log N)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d>0. This theorem is a two-dimensional generalization of…

We establish asymptotic upper bounds on the number of zeros modulo $p$ of certain polynomials with integer coefficients, with $p$ prime numbers arbitrarily large. The polynomials we consider have degree of size $p$ and are obtained by…

We answer a number of questions of Erd\H{o}s on the existence of arithmetic progressions in $k$-full numbers (i.e. integers with the property that every prime divisor necessarily occurs to at least the $k$-th power). Further, we deduce a…

We prove that many representations $\overline{\rho} : \operatorname{Gal}(\overline{K} / K) \to \operatorname{GL}_2(\mathbb{F}_3)$, where $K$ is a CM field, arise from modular elliptic curves. We prove similar results when the prime $p = 3$…

Let $K$ be a number field defined by a monic irreducible polynomial $F(X) \in \mathbb{Z}[X]$, $p$ a fixed rational prime, and $\nu_p$ the discrete valuation associated to $p$. Assume that $\overline{F}(X)$ factors modulo $p$ into the…

We prove that there exist infinitely many coprime numbers $a$, $b$, $c$ with $a+b=c$ and $c>\operatorname{rad}(abc)\exp(6.563\sqrt{\log c}/\log\log c)$. These are the most extremal examples currently known in the $abc$ conjecture, thereby…

In this paper we deal with composite rational functions having zeros and poles forming consecutive elements of an arithmetic progression. We also correct a result published earlier related to composite rational functions having a fixed…

If p is a prime, then the numbers 1, 2, ..., p-1 form a group under multiplication modulo p. A number g that generates this group is called a primitive root of p; i.e., g is such that every number between 1 and p-1 can be written as a power…

In this article we consider the completely multiplicative sequences $(a_n)_{n \in \mathbf{N}}$ defined on a field $\mathbf{K}$ and satisfying $$\sum_{p| p \leq n, a_p \neq 1, p \in \mathbf{P}}\frac{1}{p}<\infty,$$ where $\mathbf{P}$ is the…

Let $\xi$ and $m$ be integers satisfying $\xi\ne 0$ and $m\ge 3$. We show that for any given integers $a$ and $b$, $b \neq 0$, there are $\frac{\varphi(m)}{2}$ reduced residue classes modulo $m$ each containing infinitely many primes $p$…

We prove that it is always possible to find a permutation $p$ on the set $\{1,...,n\}$ such that $c+p(c)$ is prime for all $c \in \{1,...,n\}.$

Let $\Z/pZ$ be the finite field of prime order $p$ and $A$ be a subsequence of $\Z/pZ$. We prove several classification results about the following questions: (1) When can one represent zero as a sum of some elements of $A$ ? (2) When can…

We show that the existence of arithmetic progressions with few primes, with a quantitative bound on "few", implies the existence of larger gaps between primes less than x than is currently known unconditionally. In particular, we derive…

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1<a<p\ \text{and}\ \left(\frac…

We show an improved upper estimate for van der Waerden number $W(3,k):$ there is an absolute constant $c>0$ such that if $\{1,\dots,N\}=X\cup Y$ is a partition such that $X$ does not contain any arithmetic progression of length $3$ and $Y$…

In this note, we review some facts about polynomials representing functions modulo primes p. In addition we prove that the polynomial f(x) = x^{p-2} + x^{p-3} + ... + x^3 + x^2 + 2x + 1 represents the transposition (0 1) modulo p, that is,…

Let $\BZ_p$ be the finite field of prime order $p$ and $A$ be a subset of $\BZ_p$. We prove several sharp results about the following two basic questions: (1) When can one represent zero as a sum of distinct elements of $A$ ? (2) When can…

In this paper we present a class of maps for which the multiplicativity of the maximal output p-norm holds when p is 2 and p is larger than or equal to 4. The class includes all positive trace-preserving maps from the matrix algebra on the…