相关论文: Simple arguments on consecutive power residues

We show that polynomial recursions $x_{n+1}=x_{n}^{m}-k$ where $k,m$ are integers and $m$ is positive have no nontrivial periodic integral orbits for $m\geq3$. If $m=2$ then the recursion has integral two-cycles for infinitely many values…

We prove that if $k$ is a positive integer then for every finite field $\mathbb{F}$ of cardinality $q\neq 2$ and for every positive integer $n$ such that $q^n>(k-1)^4$, every $n\times n$ matrix over $\mathbb{F}$ can be expressed as a sum of…

We continue the study of positive energy (lowest weight) unitary irreducible representations of the superalgebras osp(1|2n,R). We present the full list of these UIRs. We give the Proof of the case osp(1|8,R).

We show, in an effective way, that there exists a sequence of congruence classes $a_k\pmod {m_k}$ such that the minimal solution $n=n_k$ of the congruence $\phi(n)\equiv a_k\pmod {m_k}$ exists and satisfies $\log n_k/\log m_k\to\infty $ as…

We give the q-analogue of the sums of the n-th powers of positive integers up to k-1.

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…

A partition of a positive integer $n$ is a non-increasing sequence of positive integers which sum to $n$. A recently studied aspect of partitions is the minimal excludant of a partition, which is defined to be the smallest positive integer…

We obtain a totient analogue for Linnik's theorem in arithmetic progressions. Specifically, for any coprime pair of positive integers $(m,a)$ such that $m$ is odd, there exists $n\le m^{2+o(1)}$ such that $\varphi(n)\equiv…

We will prove several congruences modulo a power of a prime such as $$ \sum_{0<k_1<...<k_{n}<p}\leg{p-k_{n}}{3} {(-1)^{k_{n}}\over k_1... k_{n}}\equiv {lll} -{2^{n+1}+2\over 6^{n+1}} p B_{p-n-1}({1\over 3}) &\pmod{p^2} &{if $n$ is odd}…

Finding the $n$-th positive square number is easy, as it is simply $n^2$. But how do we find the complementary sequence, i.e., the $n$-th positive non-square number? For this case there is an explicit formula. However, for general…

We generalize the definition and properties of root systems to complex reflection groups - roots become rank one projective modules over the ring of integers of a number field k. In the irreducible case, we provide a classification of root…

Using GCD sums, we show that the set of the primes has small common multiplicative energy with an arbitrary exponentially big integer set $S$ and, in particular, size of any arithmetic progression in $S$ having the beginning at zero, is at…

Let $a > 1$. Then $a^n < n!$ for some positive integer $n$. There are several numerical sequences associated with the study of the smallest such integer which are studied in \cite{RadFact} and \cite{RadGamma}. Here we continue the…

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a…

We show that if p is an odd prime then $$\sum_{k=0}^{p-1}E_kE_{p-1-k}=1 (mod p)$$ and $$\sum_{k=0}^{p-3}E_kE_{p-3-k}=(-1)^{(p-1)/2}2E_{p-3} (mod p),$$ where E_0,E_1,E_2,... are Euler numbers. Moreover, we prove that for any positive integer…

Let $\rho_k(N)$ denote the maximum size of a set $A\subseteq \{1,2,\dots,N\}$ such that no product of $k$ distinct elements of $A$ is a perfect $d$-th power. In this short note, we prove that $\rho _d(N)=\sum\limits_{k=1}^{d-1}\pi\left(…

We prove results that imply, under various hypotheses, that every elliptic curve over a number field $k$ corresponding to a point on a modular curve has bad reduction at a certain prime $p$ of $\mathcal{O}_k$. For example, every elliptic…

Let $\mathbb{F}_{q}$ be a finite field with $q$ elements and $\mathbb{F}_{q}[x]$ the ring of polynomials over $\mathbb{F}_{q}$. Let $l(x), k(x)$ be coprime polynomials in $\mathbb{F}_{q}[x]$ and $\Phi(k)$ the Euler function in…

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive…

We present counting methods for some special classes of multivariate polynomials over a finite field, namely the reducible ones, the s-powerful ones (divisible by the s-th power of a nonconstant polynomial), and the relatively irreducible…