Related papers: A Universal Quaternary Quadratic Form over Gaussia…

Let $f$ be a positive definite (non-classic) integral quaternary quadratic form. We say $f$ is strongly $s$-regular if it satisfies a regularity property on the number of representations of squares of integers. In this article, we prove…

A positive quadratic form is $(k,\ell)$-universal if it represents all the numbers $kx+\ell$ where $x$ is a non-negative integer, and almost $(k,\ell)$-universal if it represents all but finitely many of them. We prove that for any $k,\ell$…

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…

Let $d>r\ge 0$ be integers. For positive integers $a,b,c$, if any term of the arithmetic progression $\{r+dn:\ n=0,1,2,\ldots\}$ can be written as $ax^2+by^2+cz^2$ with $x,y,z\in\mathbb{Z}$, then the form $ax^2+by^2+cz^2$ is called…

A (positive definite and integral) quadratic form $f$ is said to be $\textit{universal}$ if it represents all positive integers, and is said to be $\textit{primitively universal}$ if it represents all positive integers primitively. We also…

Every quadratic form represents 0; therefore, if we take any number of quadratic forms and ask which integers are simultaneously represented by all members of the collection, we are guaranteed a nonempty set. But when is that set more than…

In this paper, we prove some results of restricted sums of four squares using arithmetic of quaternions in the ring of Lipschitz integers. For example, we show that every nonnegative integer $n$ can be written as $x^{2}+y^{2}+z^{2}+t^{2}$…

We provide explicit formulas for quadratic Gauss sums over $\mathbb{Z}^n/c\mathbb{Z}^n$, which generalize some of the existing formulas, e.g., Skoruppa and Zagier's (for $n=2$), and Iwaniec and Kowalski's (for arbitrary $n$). We then give…

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive…

Following Bhargava and Hanke's celebrated 290-theorem, we prove a universality theorem for all positive-definite integer-valued quadratic forms that represent all positive integers coprime to $3$. In particular, if a positive-definite…

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer $N$, first we give the classification of positive…

The classical theorems relating integral binary quadratic forms and ideal classes of quadratic orders have been of tremendous importance in mathematics, and many authors have given extensions of these theorems to rings other than the…

In this paper, we find a basis for the space of modular forms of weight $2$ on $\Gamma_1(48)$. We use this basis to find formulas for the number of representations of a positive integer $n$ by certain quaternary quadratic forms of the form…

This note presents a short, transparent proof of the theorem that every Euclidean quadratic form over a normed integral domain is an Aubry-Davenport-Cassels form. The theorem, as formulated in the note, allows besides quadratic terms also…

In 2016, while studying restricted sums of integral squares, Sun posed the following conjecture: Every positive integer $n$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\in\mathbb{N}=\{0,1,\cdots\})$ with $x+3y$ a square. Meanwhile, he also…

A proof of Lagrange's and Jacobi's four-square theorem due to Hurwitz utilizes orders in a quaternion algebra over the rationals. Seeking a generalization of this technique to orders over number fields, we identify two key components: an…

In 1997 K. Ono and K. Soundararajan [Invent. Math. 130(1997)] proved that under the generalized Riemann hypothesis any positive odd integer greater than 2719 can be represented by the famous Ramanujan form $x^2+y^2+10z^2$, equivalently the…

We investigate here the representability of integers as sums of triangular numbers, where the $n$-th triangular number is given by $T_n = n(n + 1)/2$. In particular, we show that $f(x_1,x_2,..., x_k) = b_1 T_{x_1} +...+ b_k T_{x_k}$, for…

Several physical systems (two identical particles in two dimensions, isotropic oscillator and Kepler system in a 2-dim curved space) and mathematical structures (quadratic algebra QH(3), finite W algebra $\bar {\rm W}_0$) are shown to…

Let $a_k(n)$ denotes the number of representations of a non-negative integer $n$ as sum of $k$ quadratic forms of the type $x^2+xy+y^2$ and $a_{\lambda_1,\lambda_2,\lambda_3\dots\lambda_k}(n)$ denotes the number of representations $n$ as a…