Related papers: Binary quadratic forms and the factorization metho…
Section 235 of Gauss' fundamental treatise "Disquisitiones Arithmeticae" establishes basic properties that compositions of binary quadratic forms must satisfy. Although this section is very technical, it contains truly important results. We…
Over 200 years ago, Gauss discovered a composition law on the $SL_2({\mathbb Z})$-equivalence classes of primitive binary quadratic forms. Since then, bijections of classes of binary forms have been found with ideal class groups of…
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…
We use the periodicity properties of generalized Gauss sums to factor numbers. Moreover, we derive rules for finding the factors and illustrate this factorization scheme for various examples. This algorithm relies solely on interference and…
The aim of this paper is to introduce and study a large class of $\mathfrak{g}$-module algebras which we call factorizable by generalizing the Gauss factorization of (square or rectangular) matrices. This class includes coordinate algebras…
Number theory as a coherent mathematical subject started with the work of Fermat in the decade from 1630 to 1640, but modern number theory, that is, the systematic and mathematically rigorous development of the subject from fundamental…
This exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and…
We apply general difference calculus in order to obtain solutions to the functional equations of the second order. We show that factorization method can be successfully applied to the functional case. This method is equivariant under the…
Several physics-based algorithms for factorizing large number were recently published. A notable recent one by Schleich et al. uses Gauss sums for distinguishing between factors and non-factors. We demonstrate two NMR techniques that…
The classical quadratic Gauss sum can be thought of as an exponential sum attached to a quadratic form on a cyclic group. We introduce an equivariant version of Gauss sum for arbitrary finite quadratic forms, which is an exponential sum…
Dirichlet's version of Gauss's reduction theory for indefinite binary quadratic forms includes a map from Gauss-reduced forms to strings of natural numbers. It attaches to a form the minimal period of the continued fraction of a quadratic…
The purpose of this note is to provide an alternative proof of two quadratic transformation formulas contiguous to that of Gauss using a differential equation approach.
A generalization of the factorization technique is shown to be a powerful algebraic tool to discover further properties of a class of integrable systems in Quantum Mechanics. The method is applied in the study of radial oscillator, Morse…
Let $1 \le k \le n,$ and let $v_1,\ldots,v_k$ be integral vectors in $\mathbb{Z}^n$. We consider the wedge map $\alpha_{n,k} : (\mathbb{Z}^n)^k /SL_k(\mathbb{Z}) \rightarrow \wedge^k(\mathbb{Z}^n)$, $(v_1,\ldots,v_k) \rightarrow v_1 \wedge…
Factorization of numbers with the help of Gauss sums relies on an intimate relationship between the maxima of these functions and the factors. Indeed, when we restrict ourselves to integer arguments of the Gauss sum we profit from a…
A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: f(x,y,t) = a(x,y,t)/((x-t)^2+y^2) + b(x,y,t)/([(x-t)^2+y^2]^{1/2}) +…
In 2001, Bhargava proved a composition law for $2 \times 2 \times 2$ integer cubes, which generalized Gauss composition of integral binary quadratic forms. Furthermore, he derived four new composition laws defined on the following spaces:…
This article is an abridged and commented translation into Spanish of the 1815 memoir where Gauss introduced the quadrature rules now associated with his name. Gauss' work does not resemble at all the stardard text-book treatment of…
An analysis of the characteristic function of Gaussian quadratic forms is presented in [1] to study the performance of multichannel communication systems. This technical report reviews this analysis, obtaining alternative expressions to…
We obtain divisibility conditions on the multiplicative orders of elements of the form $\zeta + \zeta^{-1}$ in a finite field by exploiting a link to the arithmetic of real quadratic fields.