Related papers: On a diophantine inequality with prime numbers of …

Let $1<c<\frac{1787}{1502}$ and $N$ be a sufficiently large real number. In this paper, it is proved that for any arbitrarily large number $E>0$ and for almost all real $R \in (N,2N]$, the Diophantine inequality…

Let $1<c<37/18,\,c\neq2$ and $N$ be a sufficiently large real number. In this paper, we prove that, for almost all $R\in(N,2N],$ the Diophantine inequality $|p_1^c+p_2^c+p_3^c-R|<\log^{-1}N$ is solvable in primes $p_1,\,p_2,\,p_3.$…

Let $1<c<\frac{26088036}{12301745},c\not=2$ and $N$ be a sufficiently large real number. In this paper, it is proved that, for almost all $R\in (N,2N]$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c-R\big|<\log^{-1}N…

Let $1<c<832/825$. For large real numbers $N>0$ and a small constant $\vartheta>0$, the inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\vartheta \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4$ such that,…

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…

In this paper we show that, for any fixed $1<c<\frac{5363}{3900}$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c+p_5^c-N|<\varepsilon…

Let $c$ be fixed with $1<c<35/34$. In this paper we prove that for every sufficiently large real number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c-N|<\varepsilon \end{equation*}…

In this paper we solve the ternary Piatetski-Shapiro inequality with prime numbers of a special form. More precisely we show that, for any fixed $1<c<\frac{427}{400}$, every sufficiently large positive number $N$ and a small constant…

We consider the equation $[p_{1}^{c}] + [p_{2}^{c}] + [p_{3}^{c}] = N$, where $N$ is a sufficiently large integer, and prove that if $1 < c < \frac{17}{16}$, then it has a solution in prime numbers $p_{1}$, $p_{2}$, $p_{3}$ such that each…

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 $[x]$ denote the integral part of the real number $x$, and $N$ be a sufficiently large integer. In this paper, it is proved that, for $1<c<\frac{4109054}{1999527}, c\not=2$, the Diophantine equation…

In this paper we show that, for any fixed $1<c<967/805$, every sufficiently large positive number $N$ and a small constant $\varepsilon>0$, the diophantine inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\varepsilon \end{equation*}…

Let $[\, x\,]$ denote the integer part of a real number $x$. Assume that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real. Let…

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…

The inequalities concern the sum of s powers of primes with non-integer exponent c>1. Here s =2,3,4,or 5. The equations are similar, taking integer part before summing; here s = 3 or 5. New ranges of c are found in all cases for which many…

Let $c > 1$ and $0 < \gamma < 1$ be real, with $c \notin \mathbb N$. We study the solubility of the Diophantine inequality \[ \left| p_1^c + p_2^c + \dots + p_s^c - N \right| < \varepsilon \] in Piatetski-Shapiro primes $p_1, p_2, \dots,…

We show that whenever $\delta>0$, $\eta$ is real and constants $\lambda_i$ satisfy some necessary conditions, there are infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality $|\lambda_1p_1 + \lambda_2p_2 +…

Let $[\,\cdot\,]$ be the floor function. In this paper we show that whenever $\eta$ is real, the constants $\lambda_i$ satisfy some necessary conditions, then for any fixed $1<c<38/37$ there exist infinitely many prime triples $p_1,\,…

We prove that there are infinitely many solutions of $$ |\lambda_0+\lambda_1p+\lambda_2P_r|<p^{-\tau}, $$ where $r=3,$ $\tau=\frac1{118}$, and $\lambda_0$ is an arbitrary real number and $\lambda_1,\lambda_2\in\BR$ with $\lambda_2\neq0$ and…

Assume that $N$ is a sufficiently large positive number. In this paper we show that for a small constant $\varepsilon>0$, the logarithmic inequality \begin{equation*} \big|p_1\log p_1+p_2\log p_2+p_3\log p_3-N\big|<\varepsilon…