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In 1992, Erd$\H{o}$s and Hegyv$\'{a}$ri showed that for any prime p, there exist infinitely many length 3 weakly prime-additive numbers divisible by p. In 2018, Fang and Chen showed that for any positive integer m, there exists infinitely…

We examine the equation given by \begin{equation} \label{eq_abstract} -\Delta u + a(x) \cdot \nabla u = u^p \qquad \mbox{in $ \IR^N$,} \end{equation} where $p>1$ and $ a(x)$ is a smooth vector field satisfying some decay conditions. We show…

In this note we prove that for every integer $d \geq 1$, there exists an explicit constant $B_d$ such that the following holds. Let $K$ be a number field of degree $d$, let $q > \max\{d-1,5\}$ be any rational prime that is totally inert in…

Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval…

Let $a, b, c,$ and $n$ be integers, with $a$ nonzero and $n$ at least two. Necessary and sufficient conditions on these parameters are derived which guarantee that all solutions of the congruence \[ ax^2+bx+c \equiv 0\ \textrm{mod}\ n \]…

We prove that if $x$ is large enough, namely $x\ge x_0$, then there exists a prime between $x(1- \Delta^{-1})$ and $x$, where $\Delta$ is an effective constant computed in terms of $x_0$. This improves some previous results of Ramar\'e and…

In scientific paper, we will show a proof of Legendre's conjecture based on a scheme for finding elements of "active" set ${\rm H}_{m^{4}}$ and "critical" element $C_{m^{4}}$ for number $m^{4}$ at each $m \ge 3$.

In this paper we obtain an asymptotic formula for the number of $\operatorname{SL}_2(\mathbb{Z})$-equivalence classes of positive definite binary quadratic forms over $\bZ$ having bounded discriminant $\Delta = 1-4p$, with $p$ a prime. We…

Let the numbers $\alpha_n,\beta_n$ and $\gamma_n$ denote \begin{align*} \alpha_n=\sum_{k=0}^{n-1}{2k\choose k},\quad \beta_n=\sum_{k=0}^{n-1}{2k\choose k}\frac{1}{k+1}\quad\text{and}\quad \gamma_n=\sum_{k=0}^{n-1}{2k\choose…

Catalan's conjecture claims that the Diophantine equation $x^p-y^q=1$ admits the unique solution $3^2-2^3=1$ in integers $x,y,p,q \ge 2$. The conjecture has been finally proved by P. Mih\u{a}ilescu (2002) using the theory of cyclotomic…

In this paper we study a Landis-type conjecture for fractional Schr\"odinger equations of fractional power $s\in(0,1)$ with potentials. We discuss both the cases of differentiable and non-differentiable potentials. On the one hand, it turns…

The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod…

We show that for each positive integer $a$ there exist only finitely many prime numbers $p$ such that $a$ appears an odd number of times in the period of continued fraction of $\sqrt{p}$ or $\sqrt{2p}$. We also prove that if $p$ is a prime…

Let $\mathbb{Z}^{+}$ be the set of positive integers. Let $C_{k}$ denote all subsets of $\mathbb{Z}^{+}$ such that neither of them contains $k + 1$ pairwise coprime integers and $C_k(n)=C_k\cap \{1,2,\ldots,n\}$. Let $f(n, k) =…

Let $\mathfrak{p}_d(q)$ denote the critical exponent of the Riesz projection from $L^q(\mathbb{T}^d)$ to the Hardy space $H^p(\mathbb{T}^d)$, where $\mathbb{T}$ is the unit circle. We present the state-of-the-art on the conjecture that…

Zaremba's conjecture (1971) states that every positive integer number d can be represented as a denominator of a finite continued fraction b/d = [d1,d2,...,dk], with all partial quotients d1,d2,...,dk being bounded by an absolute constant…

This paper and [29] treat the existence and nonexistence of stable weak solutions to a fractional Hardy--H\'enon equation $(-\Delta)^s u = |x|^\ell |u|^{p-1} u$ in $\mathbb{R}^N$, where $0 < s < 1$, $\ell > -2s$, $p>1$, $N \geq 1$ and $N >…

We prove that for every irrational number $\alpha$, real number $\beta$, real number $c$ satisfying $1<c<9/8$ and positive real number $\theta$ satisfying $\theta<(9/c-8)/10$, there exist infinitely many primes of the form…

In this paper, we study the existence and multiplicity of solutions for the following fractional problem involving the Hardy potential and concave-convex nonlinearities: $$({-}{ \Delta})^{\frac{\alpha}{2}}u- \gamma \frac{u}{|x|^{\alpha}}=…

An outstanding conjecture on roots of Ehrhart polynomials says that all roots $\alpha$ of the Ehrhart polynomial of an integral convex polytope of dimension $d$ satisfy $-d \leq \Re(\alpha) \leq d-1$. In this paper, we suggest some…