Related papers: Diophantine approximation by Piatetski-Shapiro pri…

Let $F \in \mathbb Z[x, y]$ be an irreducible binary form of degree $d \geq 7$ and content one. Let $\alpha$ be a root of $F(x, 1)$ and assume that the field extension $\mathbb Q(\alpha)/\mathbb Q$ is Galois. We prove that, for every…

Let $${\mathbb P}^c=(\lfloor p^c\rfloor)_{p\in{\mathbb P}} \qquad (c>1,\ c\not\in {\mathbb N}), $$ where ${\mathbb P}$ is the set of prime numbers, and $\lfloor\cdot\rfloor$ is the floor function. We show that for every such $c$ there are…

A sequence of integers of the form $\lfloor n^{\alpha}\rfloor$ $(n=1,2,\ldots)$ for some fixed non-integral $\alpha>1$ is called a Piatetski-Shapiro sequence, where $\lfloor x\rfloor$ denotes the integer part of $x$. Let…

Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$ such that $$ \{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\} $$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).

Let $\{u_{n}\}_{n \geq 0}$ be a non-degenerate binary recurrence sequence with positive, square-free discriminant and $p$ be a fixed prime number. In this paper, we have shown the finiteness result for the solutions of the Diophantine…

For $\alpha>1$, set $\beta=1/(\alpha-1)$. We show that, for every $1<\alpha<(\sqrt{21}+4)/5\approx1.717$, the number of pairs $(m,n)$ of positive integers with $d=\lfloor{n^\alpha}\rfloor - \lfloor{m^\alpha}\rfloor$ is equal to…

Let $\mathfrak{p}=(\mathfrak{p}_1,...,\mathfrak{p}_r)$ be a system of $r$ polynomials with integer coefficients of degree $d$ in $n$ variables $\mathbf{x}=(x_1,...,x_n)$. For a given $r$-tuple of integers, say $\mathbf{s}$, a general local…

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not…

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical…

Let $F_{n}$ be the $n$-th Fibonacci number. Put $\varphi=\frac{1+\sqrt5}{2}$. We prove that the following inequalities hold for any real $\alpha$: 1) $\inf_{n \in \mathbb N} ||F_n\alpha||\le\frac{\varphi-1}{\varphi+2}$, 2) $\liminf_{n\to…

For any real number $t$, let $[t]$ denote the integer part of $t$. In this paper it is proved that if $1<c<\frac{247}{238}$, then for sufficiently large integer $N$, the equation \[\left[p^{c}\right]+\left[m^{c}\right]=N\] has a solution in…

We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n\}_{n\in\mathbb{N}},\,…

In the present paper, we have developed a method for solving \textit{diophantine inequalities} using their relationship with the \textit{difference between consecutive primes}. Using this approach we have been able to prove some theorems,…

In a previous paper, the authors proved that in any system of three linear forms satisfying obvious necessary local conditions, there are at least two forms that infinitely often assume $E_2$-values; i.e., values that are products of…

In this paper, we prove that for any fixed $205/243<\gamma\leqslant1$, every sufficiently large $N$ satisfying $N\equiv 5 \pmod {24}$ can be represented as five squares of primes with one prime in $\mathcal{P}_\gamma$, which improves the…

For any given positive definite binary quadratic form $Q$ with integer coefficients, we establish two results on Diophantine approximation with integers represented by $Q$. Firstly, we show that for every irrational number $\alpha$, there…

Infinite exponential sequences of distinct prime numbers of the form $\lfloor a c^{n^d}+b\rfloor$, $n\geq 0$, are proved to exist for well chosen real constants $a>0$, $b$, $c>1$, $d>1$, assuming Cramer's conjecture on prime gaps. There is…

By using Beta Dirichlet series and then Eisenstein series we ca represent primes with first a good approximation and an exact expression. This can be done with arbitrary prime (up to 10^101).

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then \[ |(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| >…

Let $\Theta=(\alpha,\beta)$ be a point in $\bR^2$, with $1,\alpha,\beta$ linearly independent over $\bQ$. We attach to $\Theta$ a quadruple $\Omega(\Theta)$ of exponents which measure the quality of approximation to $\Theta$ both by…