Related papers: Large gaps between values of several binary quadra…

Let $d_n = p_{n+1} - p_n$, where $p_n$ denotes the $n$th smallest prime, and let $R(T) = \log T \log_2 T\log_4 T/(\log_3 T)^2$ (the "Erd{\H o}s--Rankin" function). We consider the sequence $(d_n/R(p_n))$ of normalized prime gaps, and show…

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree $ \theta ,$ where $ \theta >2$ is real and non-integral. For fixed non-zero real numbers $ \lambda_i $ not all of the same sign we write…

Given a positive definite binary quadratic form f, let r(n) = |{(x,y): f(x,y)=n}| denote its representation function. In this paper we study linear correlations of these functions. For example, if r_1, ..., r_k are representation functions,…

We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.

We prove that every sufficiently large integer $n$ can be written in the form $n=x^2+y^2-z^2$ with $\textrm{max}(x^2,y^2,z^2)\le n$. The proof converts the problem into finding a primitive binary quadratic form of positive discriminant $4n$…

The aim of this note is to provide an upper bound of the number of positive integers $\le x$ which can be written as $\varphi(n)$ for some positive integer $n$, where $\varphi$ stands for the Euler's function. The order of magnitude of this…

We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in…

For each integer $d\ge 4$, we study the sequence of positive integers which are represented by one at least of the cyclotomic binary forms $\Phi_n(X,Y)$, with $n$ a positive integer satisfying $\varphi(n)\ge d$. The case $d=2$ was studied…

Motivated by a question of S\'ark\"ozy, we study the gaps in the product sequence $\B=\A ... \A=\{b_n=a_ia_j, a_i,a_j\in \A\}$ when $\A$ has upper Banach density $\alpha>0$. We prove that there are infinitely many gaps $b_{n+1}-b_n\ll…

In this paper, we consider the following diagonal quadratic forms \begin{equation*} a_1x_1^2 + a_2x_2^2 + \cdots + a_{\ell}x_{\ell}^2, \end{equation*} where $\ell\ge 5$ is an odd integer and $a_i\ge 1$ are integers. By using the extended…

Let $D<0$ be a fundamental discriminant and $h(D)$ be the class number of $\mathbb{Q}(\sqrt{D})$. Let $R(X,D)$ be the number of classes of the binary quadratic forms of discriminant $D$ which represent a prime number in the interval…

Let $Q$ be a positive-definite quaternary quadratic form with prime discriminant. We give an explicit lower bound on the number of representations of a positive integer $n$ by $Q$. This problem is connected with deriving an upper bound on…

This note presents a result on the maximal prime gap of the form p_(n+1) - p_n <= C(log p_n)^(1+e), where C > 0 is a constant, for any arbitrarily small real number e > 0, and all sufficiently large integer n > n_0. Equivalently, the result…

Let $k\geq 2$ be a fixed natural number. We establish the existence of infinitely many pairs of consecutive primes $p_n$, $p_{n+1}$ satisfying $$ p_{n+1}-p_n\geq c\:\frac{\log p_n\: \log_2 p_n\: \log_4 p_n}{\log_3 p_n}\:,$$ with $c$ being a…

We use Maynard's methods to show that there are bounded gaps between primes in the sequence $\{\lfloor n\alpha\rfloor\}$, where $\alpha$ is an irrational number of finite type. In addition, given a superlinear function $f$ satisfying some…

Let $\mathcal{L}$ be the closure of the set of all real numbers $\alpha$, such that there exist infinitely many integers $n$, such that $\alpha=\log\frac{d(n+1)}{d(n)}$, where $d$ is the number of divisors of $n$. We give improved lower…

We describe various properties of continued fraction expansions of complex numbers in terms of Gaussian integers. Numerous distinct such expansions are possible for a complex number. They can be arrived at through various algorithms, as…

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…

We refine a remark of Steinerberger (2024), proving that for $\alpha \in \mathbb{R}$, there exists integers $1 \leq b_{1}, \ldots, b_{k} \leq n$ such that \[ \left\| \sum_{j=1}^k \sqrt{b_j} - \alpha \right\| = O(n^{-\gamma_k}), \] where…

We study conditions on $f$ under which an $f$-divergence $D_f$ will satisfy $D_f \geq c_f V^2$ or $D_f \geq c_{2,f} V^2 + c_{4,f} V^4$, where $V$ denotes variational distance and the coefficients $c_f$, $c_{2,f}$ and $c_{4,f}$ are {\em best…