Related papers: Diophantine approximation by Piatetski-Shapiro pri…

Suppose that $c,d,\alpha,\beta$ are real numbers satisfying the inequalities $1<d<c<79/71$ and $1<\alpha<\beta<6^{1-d/c}$. In this paper, it is proved that, for sufficiently large real numbers $N_1$ and $N_2$ subject to $\alpha\leqslant…

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $p\alpha$ when $\alpha$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the…

In this paper we introduce a new diophantine inequality with prime numbers. Let $1<c<\frac{10}{9}$. We show that for any fixed $\theta>1$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the tangent…

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists).…

We introduce a method for showing that there exist prime numbers which are very close together. The method depends on the level of distribution of primes in arithmetic progressions. Assuming the Elliott-Halberstam conjecture, we prove that…

We prove a strong simultaneous Diophantine approximation theorem for values of additive and multiplicative functions provided that the functions have certain regularity on the primes.

The author proves that for $0.9985 < \gamma < 1$, there exist infinitely many primes $p$ such that $[p^{1/\gamma}]$ has at most 5 prime factors counted with multiplicity. This gives an improvement upon the previous results of…

Assume that $\lambda_1, \lambda_2, \lambda_3,\lambda_4,\lambda_5,\lambda_6,\lambda_7$ are non-zero real numbers , $\lambda_1/\lambda_2$ is an irrational number. Let $\mathcal{V} $ be a well-spaced sequence, and $\delta >0$. For any given…

Let $\Theta = (\theta_1,\theta_2,\theta_3)\in \mathbb{R}^3$. Suppose that $1,\theta_1,\theta_2,\theta_3$ are linearly independent over $\mathbb{Z}$. For Diophantine exponents $$ \alpha(\Theta) = \sup \{\gamma >0:\,\,\, \limsup_{t\to…

In this note we consider the title Diophantine equation from both theoretical as well as experimental point of view. In particular, we prove that for $k=4, 6$ and each choice of the signs our equation has infinitely many co-prime positive…

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes $p=[n^{1/\gamma}]$ with…

Let $\mathbf{P}$ denote the set of prime numbers and, for an appropriate function $h$, define a set $\mathbf{P}_{h}=\{p\in\mathbf{P}: \exists_{n\in\mathbb{N}}\ p=\lfloor h(n)\rfloor\}$. The aim of this paper is to show that every subset of…

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…

We refine a result of the last two Authors of [8] on a Diophantine approximation problem with two primes and a $k$-th power of a prime which was only proved to hold for $1<k<4/3$. We improve the $k$-range to $1<k\le 3$ by combining Harman's…

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{1193}{889}$, the following Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<\log^{-1}N \end{equation*} is solvable in…

Let $\alpha\in \mathbb{R}\setminus\mathbb{Q}$ and $\beta\in \mathbb{R}$ be given. Suppose that $a_1,\ldots,a_s$ are distinct positive integers that do not contain a reduced residue system modulo $p^2$ for any prime $p$. We prove that there…

The sequence $$ \mathbb{P}^{(c)}=(\lfloor p^c \rfloor)_{p\in \mathbb{P}}\quad (c>0,c\notin \mathbb{N}), $$ is an important subsequence of the well-known Piatetski-Shapiro sequence, where $\mathbb{P}$ is the set of prime numbers and $\lfloor…

For each of the functions $f \in \{\phi, \sigma, \omega, \tau\}$ and every natural number $k$, we show that there are infinitely many solutions to the inequalities $f(p_n-1) < f(p_{n+1}-1) < \dots < f(p_{n+k}-1)$, and similarly for…

Let $[\, \cdot\,]$ be the floor function. In this paper, we prove that when $1<c<\frac{16559}{15276}$, then every sufficiently large positive integer $N$ can be represented in the form \begin{equation*} N=[p^c_1]+[p^c_2]+[p^c_3]\,,…

We prove that, for any $c_1,c_2,c_3\in(1,41/35)$, every sufficiently large odd number $N$ can be represented as the sum of three primes $N = p_1 + p_2 +p_3$ such that $p_i = \lfloor n_{i}^{c_i}\rfloor$ for some $n_i \in{\mathbb N}$ for each…