Related papers: Primitively $2$-universal senary integral quadrati…

In this paper we generalize the idea of "essentially unique" representations by ternary quadratic forms. We employ the Siegel formula, along with the complete classification of imaginary quadratic fields of class number less than or equal…

Let $G$ be a finite group and $K$ a finite field of characteristic $2$. Denote by $t$ the $2$-rank of the commutator factor group $G/G'$ and by $s$ the number of self-dual simple $KG$-modules. Then the Witt group of equivariant quadratic…

Using the methods developed for the proof that the 2-universality criterion is unique, we partially characterize criteria for the n-universality of positive-definite integer-matrix quadratic forms. We then obtain the uniqueness of Oh's…

D.H. Lehmer found a quadratic polynomial such that 326 is a primitive root for the first 206 primes represented by this polynomial. It is shown that this is related to the class number one problem and prime producing quadratics. An…

Given a totally real number field $F$, we show that there are only finitely many totally real extensions of $K$ of a fixed degree that admit a universal quadratic form defined over $F$. We further obtain several explicit classification…

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…

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…

In this article, we consider the rank of universal $m$-gonal forms for all sufficiently large $m$. Especially, we determine the minimal rank of universal $m$-gonal form and the maximal rank of kinds of proper universal $m$-gonal form.

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a…

We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms…

A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them increased by 1 is a perfect square. The first rational Diophantine quadruple was found by Diophantus, while Euler proved that there are…

An integer of the form $P_m(x)= \frac{(m-2)x^2-(m-4)x}{2}$ for an integer $x$, is called a generalized $m$-gonal number. For positive integers $\alpha_1,\dots,\alpha_u$ and $\beta_1,\dots,\beta_v$, a mixed sum…

This paper examines with elementary proofs some interesting properties of numbers in the binary quadratic form $a^2+ab+b^2$, where $a$ and $b$ are non-negative integers. Key findings of this paper are (i) a prime number $p$ can be…

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 prove that a pair of integral quadratic forms in 5 or more variables will simultaneously represent "almost all" pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity…

In this paper we use the theory of modular forms to find formulas for the number of representations of a positive integer by certain class of quadratic forms in eight variables, viz., forms of the form $a_1x_1^2 + a_2 x_2^2 + a_3 x_3^2 +…

A (positive definite and integral) quadratic form is called {\it an isolation} of a quadratic form $f$ if it represents all subforms of $f$ except for $f$ itself. The minimum rank of isolations of a quadratic form $f$ is denoted, if it…

Let $\mathbb{F}_q$ be the finite field of $q$ elements, and let $k\mid q-1$ be a positive integer. Let $f(x)=ax^2+bx+c$ be a quadratic polynomial in $\mathbb{F}_q[x]$ with $b^2-4ac\ne0$. In this paper, we show that if…

We establish an upper bound on the number of real multiquadratic fields that admit a universal quadratic lattice of a given rank, or contain a given amount of indecomposable elements modulo totally positive units, obtaining density zero…

Let $A$ be a definite quaternion algebra over $\mathbb Q$, with discriminant $D_A$, and $O$ a maximal order of $A$. We show that the minimum of the positive definite hamiltonian binary forms over $O$ with discrimiminant $-1$ is…