Related papers: Integral basis of pure fields with square-free par…

We propose an algorithm for finding zero divisors in quaternion algebras over quadratic number fields, or equivalently, solving homogeneous quadratic equations in three variables over $\mathbb{Q}(\sqrt{d})$ where $d$ is a square-free…

In 2012 the first named author conjectured that totally real quartic fields of fundamental discriminant are determined by the isometry class of the integral trace zero form; such conjecture was based on computational evidence and the analog…

For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class…

For a large integer $m,$ we obtain an asymptotic formula for the number of solutions of a certain congruence modulo $m$ with four variables, where the variables belong to special sets of residue classes modulo $m.$ This formula are applied…

We determine the squarefree part of the scalar factor that arises when the quartic invariant of the generic binary form $F$ of odd degree $2n+1$ is expressed as the discriminant of the unique quadratic covariant $(F,F)_{2n}$. This…

Let f\in Z[x], deg(f)=3. Assume that f does not have repeated roots. Assume as well that, for every prime q, the inequality f(x)\not\equiv 0 mod q^2 has at least one solution in (Z/q^2 Z)^*. Then, under these two necessary conditions, there…

We investigate the class number one problem for a parametric family of real quadratic fields of the form $\mathbb{Q}( \sqrt{m^2+4r})$ for certain positive integers $m$ and $r$.

We determine all integers $n$ such that $n^2$ has at most three base-$q$ digits for $q \in \{2, 3, 4, 5, 8, 16 \}$. More generally, we show that all solutions to equations of the shape $$ Y^2 = t^2 + M \cdot q^m + N \cdot q^n, $$ where $q$…

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ are themselves homogeneous polynomials of some degree $d$ in $r+1$ variables. Equivalently, we are looking at rational…

Let $l$ be a rational prime greater than or equal to $3$ and $k$ be a given positive integer. Under a conjecture due to Langland and an assumption on upper bound for the regulator of fields of the form $\mathbb{Q}\left(\sqrt[l]a\right)$, we…

The subject matter of this work is quadratic and cubic polynomial functions with integer coefficients;and all of whose roots are integers. The material of this work is directed primarily at educators,students,and teachers of…

Let $K = \Q(\theta)$ be an algebraic number field with $\theta$ satisfying an irreducible polynomial $x^{9} - a$ over the field $\Q$ of rationals and $\Z_K$ denote the ring of algebraic integers of $K$. In this article, we provide the exact…

A linear quaternionic equation in one quaternionic variable q is of the form $a_1 q b_1+a_2 q b_2+ ... +a_m q b_m = c$, where the $a_i, b_j, c$ are given quaternionic coefficients. If introducing basis elements $\bf i, j, k$ of pure…

In this paper, we investigate the conditions under which a diagonal quadratic form $\sum_{i=1}^{m}a_i X_i^2$ represents every $n \times n$ integral matrix, where $a_i$ ($1 \leq i \leq m$) are integers. For $n=2$, we give a necessary and…

In totally real number fields, we characterize the rings of $S$-integers (obtained by inverting a rational integer $m$) such that all their totally positive elements are represented as a sum of squares. We further obtain partial answers to…

In this paper we study a geometric coding algorithm for indefinite binary quadratic forms Q for the congruence subgroup \Gamma^0(N), with respect to the usual fundamental domain FN, where N is assumed prime. The cycles Q_1, . . ., Q_n that…

For square-free positive integers $n$, we study the action of the modular group $\mbox{PSL}(2,\mathbb{Z})$ on the subsets $\{\,\frac{a+\sqrt{-n}}{c}\in \mathbb{Q}(\sqrt{-n})\, | \, a,b=\frac{a^2+n}{c},c \in \mathbb{Z} \,\}$ of the imaginary…

Let $K$ be a totally real number field with Galois closure $L$. We prove that if $f \in \mathbb Q[x_1,...,x_n]$ is a sum of $m$ squares in $K[x_1,...,x_n]$, then $f$ is a sum of \[4m \cdot 2^{[L: \mathbb Q]+1} {[L: \mathbb Q] +1 \choose…

A $\textit{square-full}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n\leqslant N$ is denoted by…

An $n$-ary integral quadratic form is a formal expression $Q(x_1,...,x_n)=\sum_{1\leq i,j\leq n}a_{ij}x_ix_j$ in $n$-variables $x_1,...,x_n$, where $a_{ij}=a_{ji} \in \mathbb{Z}$. We present a poly$(n,k, \log p, \log t)$ randomized…