Related papers: Gaps between quadratic forms

Let s(n) be the number of representations of n as the sum of three squares. We prove a remarkable new identity for s(p^2n)- ps(n) with p being an odd prime. This identity makes nontrivial use of ternary quadratic forms with discriminants…

We study the Goldbach problem for primes represented by the polynomial $x^2+y^2+1$. The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even…

We prove that every sufficiently large odd integer is a sum of two positive squares and a prime. Let R(n) be the number of representations n = x^2 + y^2 + p with x, y >= 1 and p prime. We show that R(n) > 0 for all odd n >= n0 and obtain…

Suppose $k$ is a positive integer. In this work, we establish formulas for for the number of representations of integers by the quadratic forms $$ x_{1}^{2}+\cdots+x_{k}^{2}+l\left(x_{k+1}^{2}+\cdots+x_{2k}^{2}\right) $$ for $l\in\{2,4\}$.

We show that there is a positive constant $c_0$ such that \[\sum_{n\le x}\mu^2(n^2+1)c_0x+O_{\varepsilon}(x^{7/12+\varepsilon})\] for any fixed $\varepsilon>0$. This improves a result of Estermann [3] from 1931, in which the error term had…

We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic…

In 1973, Calder\'{o}n proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of ${3m(m+1) \over 2}$ squares of quadratic forms. Very recently, by applying Hilbert's theorem on ternary…

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. We focus attention on the problem of representing large…

Recently, Jha (arXiv:2007.04243, arXiv:2011.11038) has found identities that connect certain sums over the divisors of $n$ to the number of representations of $n$ as a sum of squares and triangular numbers. In this note, we state a…

Let $R_{m, \mathrm{sq-full}}(N)$ be a representation function for the sum of a prime and a square-full number. In this article, we prove an asymptotic formula for the sum of $R_{m, \mathrm{sq-full}}(N)$ over positive integers $N$ in a short…

Recently, the authors showed that for every irrational number $\alpha$, there exist infinitely many positive integers $n$ represented by any given positive definite binary quadratic form $Q$, satisfying $||\alpha n||<n^{-(1/2-\varepsilon)}$…

We study short intervals which contain an ``almost square'', an integer $n$ that can be factored as $n = ab$ with $a$, $b$ close to $\sqrt{n}$. This is related to the problem on distribution of $n^2 \alpha \pmod 1$ and the problem on gaps…

This article proves that there are infinitely many primes of the form a^2 + b^4, in fact getting the asymptotic formula. The main result is that \sum_{a^2 + b^4\le x} \Lambda(a^2 + b^4) = 4\pi^{-1}\kappa x^{3/4} (1 + O(\log\log x / \log x))…

H. J. S. Smith proved Fermat's two-square theorem using the notion of palindromic continuants. In this paper we extend Smith's approach to proper binary quadratic form representations in some commutative Euclidean rings, including rings of…

The main purpose of this note is to provide an elementary discussion of some simple triangles of integer numbers in particular through their connections with representation theory of $sl_2$. The triangles under consideration are the Catalan…

We prove an explicit analogue of Legendre's conjecture for almost primes. Namely, for every integer $n \geq 1$, the interval $(n^2,(n+1)^2)$ contains an integer having at most $3$ prime factors, counted with multiplicity. This improves the…

We explain the triangular gaps observed experimentally in the most popular sizes of the $h$-fold iterated sumset, $hA,$ when $A$ is a randomly chosen four-element subset of the first $q$ natural numbers, for $q$ much larger than $h.$

For finite sets A and B in the plane, we write A+B to denote the set of sums of the elements of A and B. In addition, we write tr(A) to denote the common number of triangles in any triangulation of the convex hull of A using the points of A…

Consider the number of integers in a short interval that can be represented as a sum of two squares. What is an estimate for the variance of these counts over random short intervals? We resolve a function field variant of this problem in…

The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set $P$ in a Cartesian square $\mathbb{F}_p^n\times\mathbb{F}_p^n$. More precisely, we perform horizontal and vertical sums…