Related papers: Quadratic forms representing all integers coprime …

We determine explicit formulas for the number of representations of a positive integer $n$ by quaternary quadratic forms with coefficients $1$, $2$, $5$ or $10$. We use a modular forms approach.

In this paper, we determine all the positive integers $a, b$ and $c$ such that every nonnegative integer can be represented as $$ f^{a,b}_c(x,y,z,w)=ax^2+by^2+c(z^2+zw+w^2) \,\, \textrm{with} \,\,x,y,z,w\in\mathbb{Z}. $$ Furthermore, we…

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…

Let $\mathfrak o$ be the ring of integers of a totally real number field. If $f$ is a quadratic form over $\mathfrak o$ and $g$ is another quadratic form over $\mathfrak o$ which represents all proper subforms of $f$, does $g$ represent…

An integral quadratic polynomial is called regular if it represents every integer that is represented by the polynomial itself over the reals and over the $p$-adic integers for every prime $p$. It is called complete if it is of the form…

A positive definite integral quadratic form is said to be almost (primitively) universal if it (primitively) represents all but at most finitely many positive integers. In general, almost primitive universality is a stronger property than…

We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is…

A number field $k$ admits a binary integral quadratic form which represents all integers locally but not globally if and only if the class number of $k$ is bigger than one. In this case, there are only finitely many classes of such binary…

For all positive integers $k$ and $N$ we prove that there are infinitely many totally real multiquadratic fields $K$ of degree $2^k$ over $\mathbb Q$ such that each universal quadratic form over $K$ has at least $N$ variables.

Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$…

A (positive definite primitive integral) quadratic form is called odd-regular if it represents every odd positive integer which is locally represented. In this paper, we show that there are at most 147 diagonal odd-regular ternary quadratic…

We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite…

We show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals $\mathbb{Q}$, then the set of totally positive integers in the extension does not have the Northcott property. In…

We study the universality of forms of degrees greater than 2 over rings of integers of totally real number fields. We show that such universal forms always exist, but cannot be characterized by any variant of the 290-Theorem of…

Let $f$ be a positive definite ternary quadratic form. We assume that $f$ is non-classic integral, that is, the norm ideal of $f$ is $\z$. We say $f$ is {\it strongly $s$-regular } if the number of representations of squares of integers by…

Earnest and Khosravani, Iwabuchi, and Kim and Park recently gave a complete classification of the universal binary Hermitian forms. We give a unified proof of the universalities of these Hermitian forms, relying primarily on Ramanujan's…

Let $a,b,c,d,e,f\in\mathbb N$ with $a\ge c\ge e>0$, $b\le a$ and $b\equiv a\pmod2$, $d\le c$ and $d\equiv c\pmod2$, $f\le e$ and $f\equiv e\pmod2$. If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with…

The numbers of representations of totally positive integers as sums of three integer squares in $\mathbf{Q}(\sqrt{3})$ and in $\mathbf{Q}(\sqrt{17})$, are studied by using Shimura lifting map of Hilbert modular forms. We show the following…

For each integer $x$, the $x$-th generalized pentagonal number is denoted by $P_5(x)=(3x^2-x)/2$. Given odd positive integers $a,b,c$ and non-negative integers $r,s$, we employ the theory of ternary quadratic forms to determine when the sum…

In this paper, we consider representations of integers as sums of generalized heptagonal numbers with a prescribed number of repeats of each heptagonal number appearing in the sum. In particular, we investigate the classification of such…