Related papers: Simple arguments on consecutive power residues

Let $n$ be a positive integer. For any integer $a$, we say that $a$ is idempotent modulo $n$ if $a^2\equiv a\pmod n$. The $n$-modular Erd\H{o}s-Burgess constant is the smallest positive integer $\ell$ such that any $\ell$ integers contain…

The partition number $\pi(K)$ of a simplicial complex $K\subset 2^{[m]}$ is the minimum integer $\nu$ such that for each partition $A_1\uplus\ldots\uplus A_\nu = [m]$ of $[m]$ at least one of the sets $A_i$ is in $K$. A complex $K$ is…

Let $p_n$ denote the $n$-th prime number, $\{q_n\}$ be a sequence of positive numbers and $x\in\mathbb{R}$. In this note we prove that the inequality $$q_n p_{n+1}^{x}-q_{n+1}p_{n}^{x}<p_{n}^{x}p_{n+1}^{x-1}, $$ holds for infinitely many…

We suggest a construction of the minimal polynomial $m_{\beta^k}$ of $\beta^k\in \mathbb F_{q^n}$ over $\mathbb F_q$ from the minimal polynomial $f= m_\beta$ for all positive integers $k$ whose prime factors divide $q-1$. The computations…

A $k$-modal sequence is a sequence of real numbers that can be partitioned into $k+1$ (possibly empty) monotone sections such that adjacent sections have opposite monotonicities. For every positive integer $k$, we prove that any sequence of…

It is well known that when a prime $p$ is congruent to 1 modulo 4, the sum of the quadratic residues equals the sum of the quadratic nonresidues. In this note we give analogous results for the case where $p$ is congruent to 3 modulo 4.

We present and analyze an algorithm to enumerate all integers $n\le x$ that can be written as the sum of consecutive $k$th powers of primes, for $k>1$. We show that the number of such integers $n$ is asymptotically bounded by a constant…

Given a prime $p\ge5$ and an integer $s\ge1$, we show that there exists an integer $M$ such that for any quadratic polynomial $f$ with coefficients in the ring of integers modulo $p^s$, such that $f$ is not a square, if a sequence…

In this paper, we study a density version of Waring's problem. We prove that a positive density subset of $k$th-powers forms an asymptotic additive basis of order $O(k^2)$ provided that the relative lower density of the set is greater than…

A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable…

Let $A$ be a finite set of integers. We show that if $k$ is a prime power or a product of two distinct primes then $$|A+k\cdot A|\geq(k+1)|A|-\lceil k(k+2)/4\rceil$$ provided $|A|\geq (k-1)^{2}k!$, where $A+k\cdot A=\{a+kb:\ a,b\in A\}$. We…

For any odd prime number $p$, let $(\cdot|p)$ be the Legendre symbol, and let $n_1(p)<n_2(p)<\cdots$ be the sequence of positive nonresidues modulo $p$, i.e., $(n_k|p)=-1$ for each $k$. In 1957, Burgess showed that the upper bound…

An open conjecture of Z.-W. Sun states that for any integer $n>1$ there is a positive integer $k\le n$ such that $\pi(kn)$ is prime, where $\pi(x)$ denotes the number of primes not exceeding $x$. In this paper, we show that for any positive…

An integer $k$ is called regular (mod $n$) if there exists an integer $x$ such that $k^2x\equiv k$ (mod $n$). This holds true if and only if $k$ possesses a weak order (mod $n$), i.e., there is an integer $m\ge 1$ such that $k^{m+1} \equiv…

Let $m$ be a square-free integer ($m\neq 0,\pm 1$). We show that the structure of the integral bases of the fields $K=Q(\sqrt[n]{m})$ are periodic in $m$. For $3\leq n\leq 9$ we show that the period length is $n^2$. We explicitly describe…

Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$,…

An integer point in a polyhedron is called irreducible iff it is not the midpoint of two other integer points in the polyhedron. We prove that the number of irreducible integer points in $n$-dimensional polytope with radius $k$ given by a…

There exists a set $A$ of positive integers such that the number of representations of a large positive integer $m$ as a sum of two elements of $A$ grows with a lower bound of order $\log m$, but for which there is no subset $D$ of $A$…

Let $\Lambda$ be a left and right noetherian ring and $\mod \Lambda$ the category of finitely generated left $\Lambda$-modules. In this paper we show the following results: (1) For a positive integer $k$, the condition that the subcategory…

Given an integer $g$ and also some given integers $m$ (sufficiently large) and $c_1,\dots, c_m$, we show that the number of all non-negative integers $n\le M$ with the property that there exist non-negative integers $k_1,\dots, k_m$ such…