Related papers: The density of primes dividing a particular non-li…

We describe the set of prime numbers splitting completely in the non-abelian splitting field of certain monic irreducible polynomials of degree three. As an application we establish some divisibility properties of the associated ternary…

Let $E/\mathbb Q$ be an elliptic curve, let $P\in E(\mathbb Q)$ be non-torsion, and let $(D_n)$ be the associated elliptic divisibility sequence. We study when a product \[ \prod_{i=1}^k D_{n_i} \] can be a $\rho$-th power, where $\rho$ is…

Given an elliptic curve $E$ defined over the rational numbers and a prime $p$ at which $E$ has good reduction, we consider the Galois deformation ring parametrizing lifts of the residual representation on the $p$-torsion group $E[p]$. For a…

Given an elliptic curve $E$ and a point $P$ in $E(\mathbb{R})$, we investigate the distribution of the points $nP$ as $n$ varies over the integers, giving bounds on the $x$ and $y$ coordinates of $nP$ and determining the natural density of…

Given an integer $k\ge2$, let $\omega_k(n)$ denote the number of primes that divide $n$ with multiplicity exactly $k$. We compute the density $e_{k,m}$ of those integers $n$ for which $\omega_k(n)=m$ for every integer $m\ge0$. We also show…

In this note we study numbers which occur as conductors of elliptic curves over Q. We show, by constructing families of elliptic curves with quadratic discriminant and invoking a theorem of Iwaniec, that this set contains infinitely many…

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$ without complex multiplication and with conductor $N$. For each positive integer $m$, the action of the absolute Galois group…

We consider the primes which divide the denominator of the x-coordinate of a sequence of rational points on an elliptic curve. It is expected that for every sufficiently large value of the index, each term should be divisible by a primitive…

Let $K$ be a number field with ring of integers $\mathcal O_K$, and let $\{f_k\}_{k\in \mathbb N}\subseteq \mathcal O_K[x]$ be a sequence of monic polynomials such that for every $n\in \mathbb N$, the composition $f^{(n)}=f_1\circ…

We show that for any fixed $\eps>0$, there are numbers $\delta>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains…

Let $E$ be an elliptic curve defined over rational field $\mathbb{Q}$ and $N$ be a positive integer. Now, $M_E(N)$ denotes the number of primes $p$, such that the group $E_p(\mathbb{F}_p)$ is of order $N$. We show that $M_E(N)$ follows…

Let $(x_n)_{n\geq0}$ be a linear recurrence of order $k\geq2$ satisfying $$x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}$$ for all integers $n\geq k$, where $a_1,\dots,a_k,x_0,\dots, x_{k-1}\in \mathbb{Z},$ with $a_k\neq0$. In [`The quotient…

In the early 2000s, Ramakrishna asked the question: For the elliptic curve $$ E: y^2 = x^3 - x, $$ what is the density of primes $p$ for which the Fourier coefficient $a_p(E)$ is a cube modulo $p$? As a generalization of this question,…

Let $w(n)$ be an additive non-negative integer-valued arithmetic function which is equal to $1$ on primes. We study the distribution of $n + w(n)$ $\pmod p$ and give a lower bound for the density of the set of numbers which are not…

For $n=1,2,3,\ldots$ let $S_n$ be the sum of the first $n$ primes. We mainly show that the sequence $a_n=\root n\of{S_n/n}\ (n=1,2,3,\ldots)$ is strictly decreasing, and moreover the sequence $a_{n+1}/a_n\ (n=10,11,\ldots)$ is strictly…

We study the properties of the sequence of words $(B_i)$, where $B_1 = 101$ and $B_{i+1} = B_i C_i$ for $i \geq 1$, where $C_i$ is $B_i$ with the first $i$ symbols removed, and the infinite binary sequence ${\bf b} = 10101101011011101…

For a given elliptic curve $E/\mathbb{Q}$, let $N_p(E)$ be the number of points on $E$ modulo $p$ for a prime of good reduction for $E$. Given integer $n$, let $G_k(E,n)$ be the number of $k$-tuples of $p_1<p_2<\ldots <p_k$ primes of good…

We present a method for finding large fixed-size primes of the form $X^2+c$. We study the density of primes on the sets $E_c = \{N(X,c)=X^2+c,\ X \in (2\mathbb{Z}+(c-1))\}$, $c \in \mathbb{N}^*$. We describe an algorithm for generating…

The set of primes where a hypergeomeric series with rational parameters is $p$-adically bounded is known by [10] to have a Dirichlet density. We establish a formula for this Dirichlet density and conjecture that it is rare for the density…

It is known that if $p>37$ is a prime number and $E/\mathbb{Q}$ is an elliptic curve without complex multiplication, then the image of the mod $p$ Galois representation $$…