Related papers: On an equation by primes with one Linnik prime

We prove that, if $m,n\geqslant 1$ and $a_1,\ldots,a_m$ are nonnegative integers, then \begin{align*} \frac{[a_1+\cdots+a_m+1]!}{[a_1]!\ldots[a_m]!}\sum^{n-1}_{h=0}q^h\prod_{i=1}^m{h\brack a_i} \equiv 0\pmod{[n]}, \end{align*} where…

By using the complete discrimination system for polynomials, we study the number of positive solutions in {\small $C[0,1]$} to the integral equation {\small $\phi (x)=\int_0^1k(x,y)\phi ^n(y)dy$}, where {\small…

In this study, we study a Josephus problem algorithm. Let $n,k$ be positive integers and $g_k(n) = \left\lfloor \frac{n}{k-1} \right\rfloor +1$, where $ \left\lfloor \ \ \right\rfloor$ is a floor function. Suppose that there exists $p$ such…

In the present work we investigate the largest possible gaps between consecutive numbers which can be written as the difference of two primes. The best known upper bounds are the same as those concerning the largest possible difference of…

Let N be a large enough natural number, A and B be subsets of {N+1, ... , 2N}. In this paper, we prove that there exists integers a, b with a belongs to A, b belongs to B such that ab=P_k^2 + O(P_k^{1-c}), where 0<c<1/2 and P_k denotes an…

For a positive definite integral ternary quadratic form $f$, let $r(k,f)$ be the number of representations of an integer $k$ by $f$. The famous Minkowski-Siegel formula implies that if the class number of $f$ is one, then $r(k,f)$ can be…

For a positive integer $n>1$ denote by $\omega(n)$ the maximal possible number $k$ of different functions $f_1,\dots,f_k:\mathbb{Z}/n\mathbb{Z}\mapsto \mathbb{Z}/n\mathbb{Z}$ such that each function $f_i-f_j,i<j$, is bijective. Recently A.…

Let $P^+(n)$ denote the largest prime of the integer $n$. Using the \begin{align*}\Psi\_{F\_1\cdots F\_t}\left(\mathcal{K}\cap[-N,N]^d,N^{1/u}\right):=\\#\left\{\mathcal{K}\in…

Exponential sums with monomials are highly related to many interesting problems in number theory and well studied by many literatures. In this paper, we consider the exponential sums with polynomials and prove a new upper bound. As an…

We prove that there is a small but fixed positive integer e such that for every prime larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|<(2+e)|S| and 2(|2S|)-2|S|+2 < p is contained in an arithmetic…

In this paper, we determine all the positive integers $a, b$ and $c$ such that every nonnegative integer can be represented as $$ f^{a,b}_c(x,y,z,w)=ax^2+by^2+c(z^2+zw+w^2) \,\, \textrm{with} \,\,x,y,z,w\in\mathbb{Z}. $$ Furthermore, we…

Let $\pi(x;\gamma_1,\gamma_2)$ denote the number of primes $p$ with $p\leqslant x$ and $p=\lfloor n^{1/\gamma_1}_1\rfloor=\lfloor n^{1/\gamma_2}_2\rfloor$, where $\lfloor t\rfloor$ denotes the integer part of $t\in\mathbb{R}$ and…

For $k=1,2,\ldots$ let $H_k$ denote the harmonic number $\sum_{j=1}^k 1/j$. In this paper we establish some new congruences involving harmonic numbers. For example, we show that for any prime $p>3$ we have…

In this note, we show that there are many infinity positive integer values of $n$ in which, the following inequality holds $$ \left\lfloor{1/2}(\frac{(n+1)^2}{\log(n+1)}-\frac{n^2}{\log n})-\frac{\log^2 n}{\log\log…

We identify pairs of positive integers $(t, d)$ with the property that the integer sequence with general term $\lfloor{n^t/d\rfloor}$ contains at most finitely many primes.

Let $N$ be a large prime and let $c > 1/4$. We prove that if $f$ is a $\pm 1$-valued completely multiplicative function, such that the exponential sums $$ S_f(a) := \sum_{1 \leq n < N} f(n) e(na/N), \quad a \pmod{N} $$ satisfy the ``Gauss…

Let $c$ be a positive odd integer and $R$ a set of $n$ primes coprime with $c$. We consider equations $X + Y = c^z$ in three integer unknowns $X$, $Y$, $z$, where $z > 0$, $Y > X > 0$, and the primes dividing $XY$ are precisely those in…

In this report we present an off-the-number-line representation of the positive integers by expressing each integer by its unique prime signature as a grid point of an infinite dimensional space indexed by the prime numbers, which we term…

A unitary divisor $c$ of a positive integer $n$ is a positive divisor of $n$ that is relatively prime to $\displaystyle{\frac{n}{c}}$. For any integer $k$, the function $\sigma_k^*$ is a multiplicative arithmetic function defined so that…

We prove that there exists a k_0>0 such that every sufficiently large odd integer n with 3\mid n can be represented as p_1+p_2+p_3, where p_1,p_2 are Chen's primes and p_3 is a prime with p_3+2 has at most k_0 prime factors.