Related papers: A Multivariable Chinese Remainder Theorem

In this article we introduce generalized projective spaces (Definitions $[2.1, 2.5]$) and prove three main theorems in two different contexts. In the first context we prove, in main Theorem $A$, the surjectivity of the Chinese remainder…

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less…

We investigate the existence of heavy columns in binary matrices with distinct rows. A column of an m x n binary matrix is called heavy if the number of ones in it is at least m/2. We introduce two recursive algorithms, A1 and A2, that…

We generalize current known distribution results on Shanks--R\'enyi prime number races to the case where arbitrarily many residue classes are involved. Our method handles both the classical case that goes back to Chebyshev and function…

The Bergelson-Leibman theorem states that if P_1, ..., P_k are polynomials with integer coefficients, then any subset of the integers of positive upper density contains a polynomial configuration x+P_1(m), ..., x+P_k(m), where x,m are…

We give a generalization of Kung's theorem on critical exponents of linear codes over a finite field, in terms of sums of extended weight polynomials of linear codes. For all i=k+1,...,n, we give an upper bound on the smallest integer m…

Given an integer $q$ and a polynomial $f\in \mathbb Z_{q}[X]$ of degree $d$ with coefficients in the residue ring $\mathbb Z_q=\mathbb Z/q\mathbb Z,$ we obtain new results concerning the number of solutions to congruences of the form…

An integer matrix $\mathbf{A}$ is $\Delta$-modular if the determinant of each $\text{rank}(\mathbf{A}) \times \text{rank}(\mathbf{A})$ submatrix of $\mathbf{A}$ has absolute value at most $\Delta$. The study of $\Delta$-modular matrices…

Let $f(x)\in\mathbb{Z}[x]$ be a nonconstant polynomial. Let $n, k$ and $c$ be integers such that $n\ge 1$ and $k\ge 2$. An integer $a$ is called an $f$-exunit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(f(a),n)=1$. In…

We find a lower bound for the number of Chen primes in the arithmetic progression $a \bmod q$, where $(a,q)=(a+2,q)=1$. Our estimate is uniform for $q \leq \log^M x$, where $M>0$ is fixed.

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…

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…

Let $f(x) = \sum\limits _{i=0}^{n} a_i x^i $ be a polynomial with coefficients from the ring $\mathbb{Z}$ of integers satisfying either $(i)$ $0 < a_0 \leq a_{1} \leq \cdots \leq a_{k-1} < a_{k} < a_{k+1} \leq \cdots \leq a_n$ for some $k$,…

Recently, a multi-channel self-reset analog-to-digital converter (ADC) system with complex-valued moduli has been proposed. This system enables the recovery of high dynamic range complex-valued bandlimited signals at low sampling rates via…

For any positive integer $m$, let $\mathbb{Z}_{m}$ be the set of residue classes modulo $m$. For $A\subseteq \mathbb{Z}_{m}$ and $\overline{n}\in \mathbb{Z}_{m}$, let representation function $R_{A}(\overline{n})$ denote the number of…

We consider the recursive equation ``x(n+1)=A(n)x(n)'' where x(n+1) and x(n) are column vectors of size k and where A(n) is an irreducible random matrix of size k x k. The matrix-vector multiplication in the (max,+) algebra is defined by…

Let $n\ge 1, e\ge 1, k\ge 2$ and $c$ be integers. An integer $u$ is called a unit in the ring $\mathbb{Z}_n$ of residue classes modulo $n$ if $\gcd(u, n)=1$. A unit $u$ is called an exceptional unit in the ring $\mathbb{Z}_n$ if…

We pose 100 new conjectures on representations involving primes or related things, which might interest number theorists and stimulate further research. Below are five typical examples: (i) For any positive integer $n$, there exists…

Let $m,n>1$ be integers and $\mathbb{P}_{n,m}$ be the point set of the projective $(n-1)$-space (defined by [2]) over the ring $\mathbb{Z}_m$of integers modulo $m$. Let $A_{n,m}=(a_{uv})$ be the matrix with rows and columns being labeled by…

We prove that a positive proportion of the intervals of any fixed scalar multiple of $\log(X)$ in the dyadic interval $[X,2X]$ contain a prime number. We also show that a positive proportion of the congruence classes modulo $q$ contain a…