Index divisibility in the orbit of 0 for integral polynomials

The density of the terms in an elliptic divisibility sequence having a fixed G.C.D. with their index

Let $\mathbf{D}=(D_{n})_{n\geq 1}$ be an elliptic divisibility sequence associated to the pair $(E,P)$. For a fixed integer $k$, we define $\mathscr{A}_{E,k}=\{n\geq 1 : \gcd(n,D_{n})=k\}$. We give an explicit structural description of $\mathscr{A}_{E,k}$. Also, we explain when $\mathscr{A}_{E,k}$ has positive asymptotic density using bounds related to the distribution of trace of Frobenius of $E$. Furthermore, we get explicit density of $\mathscr{A}_{E,k}$ using the Möbius function.

分圆多项式的可整除性判据与系数公式

Divisibility criteria and coefficient formulas for cyclotomic polynomials

本文给出多项式被分圆多项式整除的充要条件，推导拉马努金和的新公式，并揭示分圆多项式系数的新性质及指数相差素因子时的递推关系。

Index Divisibility in Dynamical Sequences and Cyclic Orbits Modulo $p$

Let $ϕ(x) = x^d + c$ be an integral polynomial of degree at least 2, and consider the sequence $(ϕ^n(0))_{n=0}^\infty$, which is the orbit of $0$ under iteration by $ϕ$. Let $D_{d,c}$ denote the set of positive integers $n$ for which $n \mid ϕ^n(0)$. We give a characterization of $D_{d,c}$ in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes $p$ have the property that the orbit of $0$ is a single $p$-cycle modulo $p$. We show that the set of such primes is finite wh…

Elliptic Fermat numbers and elliptic divisibility sequence

For a pair $(E,P)$ of an elliptic curve $E/\mathbb{Q}$ and a nontorsion point $P\in E(\mathbb{Q})$, the sequence of \emph{elliptic Fermat numbers} is defined by taking quotients of terms in the corresponding elliptic divisibility sequence $(D_{n})_{n\in\mathbb{N}}$ with index powers of two, i.e. $D_{1}$, $D_{2}/D_{1}$, $D_{4}/D_{2}$, etc. Elliptic Fermat numbers share many properties with the classical Fermat numbers, $F_{k}=2^{2^k}+1$. In the present paper, we show that for magnified elliptic Fermat sequences, only finitely many terms are prime. We also define \emph{generalized elliptic Ferma…

Positivity and divisibility of alternating descent polynomials

The alternating descent statistic on permutations was introduced by Chebikin as a variant of the descent statistic. We show that the alternating descent polynomials on permutations are unimodal via a five-term recurrence relation. We also found a quadratic recursion for the alternating major index $q$-analog of the alternating descent polynomials. As an interesting application of this quadratic recursion, we show that $(1+q)^{\lfloor n/2\rfloor}$ divides $\sum_{π\in\mathfrak{S}_n}q^{\rm{altmaj}(π)}$, where $\mathfrak{S}_n$ is the set of all permutations of $\{1,2,\ldots,n\}$ and $\rm{altmaj}(π…

Terms in elliptic divisibility sequences divisible by their indices

Let D = (D_n)_{n\ge1} be an elliptic divisibility sequence. We study the set S(D) of indices n satisfying n | D_n. In particular, given an index n in S(D), we explain how to construct elements nd in S(D), where d is either a prime divisor of D_n, or d is the product of the primes in an aliquot cycle for D. We also give bounds for the exceptional indices that are not constructed in this way.