Related papers: Asymptotic Distribution of Residues in Pascal's Tr…

An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue…

We discuss arithmetic questions related to the "poor man's ad\`ele ring" $\mathcal A$ whose elements are encoded by sequences $(t_p)_p$ indexed by prime numbers, with each $t_p$ viewed as a residue in $\mathbb Z/p\mathbb Z$. Our main…

The nth row of Pascal's trinomial triangle gives coefficients of (1+x+x^2)^n. Let g(n) denote the number of such coefficients that are odd. We review Moshe's algorithm for evaluating asymptotics of g(n) -- this involves computing the…

For A,epsilon>0 and any sufficiently large odd n we show that for almost all k up to n^{1/5-epsilon} there exists a representation n=p1+p2+p3 with primes in residue classes b1,b2,b3 mod k for almost all admissible triplets b1,b2,b3 of…

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpi\'nski gasket that…

For a fixed rational number g, not equal to -1,0 or 1 and integers a and d we consider the set of primes p for which the order of g(mod p) is congruent to a(mod d). For d=4 and d=3 it is shown that, under the Generalized Riemann Hypothesis,…

For any prime $p$, let $y(p)$ denote the smallest integer $y$ such that every reduced residue class $\pmod p$ is represented by the product of some subset of $\{1,\dots,y\}$. It is easy to see that $y(p)$ is at least as large as the…

Let $C_n$ be the $n$th Catalan number. We show that the asymptotic density of the set $\{n: C_n \equiv 0 \mod p \}$ is $1$ for all primes $p$, We also show that if $n = p^k -1$ then $C_n \equiv -1 \mod p$. Finally we show that if $n \equiv…

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 $(p_n)$ denote the sequence of prime numbers, with $2=p_1<p_2<\ldots$. We demonstrate the existence of an irrational number $\alpha$ having the property that the sequence $(\alpha p_n)$ is not well-distributed modulo $1$.

Let $p$ be a sufficiently large prime number, $n$ be a positive odd integer with $n|\,p-1$ and $n>p^\varepsilon $, where $\varepsilon$ is a sufficiently small constant. Let $k(p,\,n)$ denote the least positive integer $k$ such that for…

Suppose that $f(x)=Ax^k$ mod $p$ is a permutation of the least residues mod $p$. With the exception of the maps $f(x)=Ax$ and $Ax^{(p+1)/2}$ mod $p$ we show that for fixed $n\geq 2$ the image of each residue class mod $n$ contains elements…

Let f(n) denote the number of odd entries in the nth row of Pascal's binomial triangle. We study "average dispersion" and "typical dispersion" of f(n) -- the latter involves computing a generalized Lyapunov exponent -- and then turn to…

We define A_n=\sum_{i=1}^n (-1)^i\frac{1}{i} and we show that, for every prime p, there exists a number n such that A_n\equiv 0 (mod p).

Let $p$ be a prime. If an integer $g$ generates a subgroup of index $t$ in $(\mathbb Z/p\mathbb Z)^*,$ then we say that $g$ is a $t$-near primitive root modulo $p$. We point out the easy result that each primitive residue class contains a…

Let $X$ be a scheme of finite type over $\mathbf{Z}$. For $p \in \mathcal{P}$ the set of prime numbers, let $N_{X}(p)$ be the number of $\mathbf{F}_{p}$-points of $X/\mathbf{F}_{p}$. For fixed $n\geq 1$ and $a_{1}, \ldots, a_{n} \in…

Let $p$ be a given modulus, let $u$ be prime to $p$, and consider the linear permutation $u\cdot n\pmod p$ of the residue system modulo $p$. Writing $\langle x\rangle_p$ to denote the least nonnegative residue of $x$ modulo $p$, we say that…

The main purpose of this paper is to present a new and more uniform model-theoretic/combinatorial proof of the theorem ([5]): The randomization $T^{R}$ of a complete first-order theory $T$ with $NIP$ is a (complete) first-order continuous…

Let $p$ be an odd prime. For any $p$-adic integer $a$ we let $\overline{a}$ denote the unique integer $x$ with $-p/2<x<p/2$ and $x-a$ divisible by $p$. In this paper we study some permutations involving quadratic residues modulo $p$. For…

Rubinstein and Sarnak investigated systems of inequalities of the form pi(x;q,a_1) > ... > pi(x;q,a_r), where pi(x;q,a) denotes the number of primes up to x that are congruent to a mod q. They showed, under standard hypotheses on the zeros…