Related papers: Distribution modulo one and denominators of the Be…

Given a separable nonconstant polynomial $f(x)$ with integer coefficients, we consider the set $S$ consisting of the squarefree parts of all the rational values of $f(x)$, and study its behavior modulo primes. Fixing a prime $p$, we…

For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial $P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k$. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related…

In this paper we study a family of polynomials $$S_n^{(m)}(x):=\sum_{i,j=0}^n\binom ni^m\binom nj^m\binom{i+j}ix^{i+j}\ \ (m,n=0,1,2,\ldots).$$ For example, we show that $$\sum_{k=0}^{p-1}S_k^{(0)}(x)\equiv\frac…

Using purely combinatorial means we obtain results on simultaneous Diophantine approximation modulo 1 for systems of polynomials with real coefficients and no constant term.

A complex polynomial $P(z) = c_0 + c_1 z +...+ c_n z^n$ is called unimodular if $|c_j| = 1$, $j = 0,...,n$. Littlewood asked the question (1966) on how close a unimodular polynomial come to satisfying $|P(z)| \approx \sqrt{n+1}$ if $n \ge…

We investigate various questions concerning the reciprocal sum of divisors, or prime divisors, of the Mersenne numbers $2^n-1$. Conditional on the Elliott-Halberstam Conjecture and the Generalized Riemann Hypothesis, we determine…

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer…

The n-th Delannoy number and the n-th Schr\"oder number given by $D_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}$ and $S_n=\sum_{k=0}^n\binom{n}{k}\binom{n+k}{k}/(k+1)$ respectively arise naturally from enumerative combinatorics. Let p be an…

Suppose that $p$ is an odd prime and $m$ is an integer not divisible by $p$. Sun and Tauraso [Adv. in Appl. Math., 45(2010), 125--148] gave $\sum_{k=0}^{n-1}\binom{2k}{k+d}/m^k$ and $\sum_{k=0}^{n-1}\binom{2k}{k+d}/(km^k)$ modulo $p$ for…

A class of self-inversive polynomials includes all the self-reciprocal polynomials. Let A denote the set of all self-reciprocal polynomials with n+1 coefficients. Let B denote the set of certain self-inversive and non self-reciprocal…

We present several congruences modulo a power of prime $p$ concerning sums of the following type $\sum_{k=1}^{p-1}{m^k\over k^r}{2k\choose k}^{-1}$ which reveal some interesting connections with the analogous infinite series.

We use recent bounds on bilinear sums with modular square roots to study the distribution of solutions to congruences $x^2 \equiv p \pmod q$ with primes $p\le P$ and integers $q \le Q$. This can be considered as a combined scenario of Duke,…

We prove that if the signed binomial coefficient $(-1)^i\binom{k}{i}$ viewed modulo p is a periodic function of i with period h prime to p in the range $0\le i\le k$, then k+1 is a power of p, provided h is not too large compared to k. (In…

By establishing an improved level of distribution we study almost primes of the form $f(p,n)$ where $f$ is an irreducible binary form over $\mathbb Z$.

We obtain asymptotic formulas for sums over arithmetic progressions of coefficients of polynomials of the form $$\prod_{j=1}^n\prod_{k=1}^{p-1}(1-q^{pj-k})^s,$$ where $p$ is an odd prime and $n, s$ are positive integers. Let us denote by…

Let $S_n=\e_1+...+\e_n$, where $ \e_i $ are i.i.d. Bernoulli r.v.'s. Let $0\le r_d(n)<2d$ be the least residue of $n$ mod$(2d)$, $\bar r_d(n)= 2d -r_d(n)$ and $\b(n,d)=\max ({1\over d}, {1\over \sqrt n})[e^{- {r_d(n)^2/2 n}} +e^{- {\bar…

In this note we show a simple formula for the coefficients of the polynomial associated with the sums of powers of the terms of an arbitrary arithmetic progression. This formula consists of a double sum involving only ordinary binomial…

The reduction modulo $p$ of a family of lacunary integer polynomials, associated with the dynamical zeta function $\zeta_{\beta}(z)$ of the $\beta$-shift, for $\beta > 1$ close to one, is investigated. We briefly recall how this family is…

Let $p$ be an odd prime, and let $\sum_{n=0}^{\infty} a_{n}X^{n}\in\mathbb{F}_p[[X]]$ be the reduction modulo $p$ of the Artin-Hasse exponential. We obtain a polynomial expression for $a_{kp}$ in terms of those $a_{rp}$ with $r<k$, for even…

In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime $p$, $$\sum_{\substack{i,j,k\ge 1\\\gcd(ijk,p)=1\\i+j+k=p}}\frac{1}{ijk}\equiv -2B_{p-3} \pmod{p},$$ where $B_n$ is the $n$-th…