Factoring isometries of quadratic spaces into reflections

Factoring Euclidean isometries

Every isometry of a finite dimensional euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this article we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimal length reflection factorizations. The model ...

Factoring in Quadratic Fields

This is closed under addition, subtraction, multiplication, and also division by nonzero numbers. Similarly, set Z[ √ d] = {a+ b √ d : a, b ∈ Z}. This is closed under addition, subtraction, and multiplication, but not usually division. We will define a concept of “integers” for K, which will play the same role in K as the ordinary integers Z do in Q. The integers of K will contain Z[ √ d] but m...

On Isometries of Euclidean Spaces

Isometries and the Linear Algebra of Quadratic

0.1. 2D. In the context of linear algebra a plane is a two-dimensional real vector space. A basis for a plane consists of any two vectors E1,E2 which span the plane. ‘Spanning’ means that any vector v⃗ in the plane can be written as v⃗ = uE1 + vE2 with u, v ∈ R. It is a theorem that for two vectors in the plane “spanning” is equivalent to being “linearly independent”. The ‘linearly independent pa...

The Quadratic Sieve Factoring Algorithm

The quadrat ic sieve algori thm i s cur ren t ly t h e method of choice t o f a c t o r very l a r g e composite numbers wi th no small fac tors . I n the hands of the Sandia Nat iona l Laboratories team of James Davis and Diane Holdridge, it has held t h e record f o r t h e l a r g e s t hard number f a c t o r e d s i n c e mid-1983. As of t h i s wri t ing, the l a r g e s t number it has c...