Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
8897043 | Journal of Number Theory | 2018 | 39 Pages |
Abstract
An old result of Berggren's says that there exist three 3Ã3 matrices N1,N2,N3 with the following remarkable property: Start with (3,4,5) or (4,3,5) and multiply N1,N2, or N3 by it in any order any number of times. This yields another primitive Pythagorean triple (x,y,z), that is, a triple of positive integers without common factor satisfying x2+y2âz2=0. Furthermore, every primitive Pythagorean triple can be obtained uniquely this way. In other words, all primitive Pythagorean triples can be given a tree-like structure with each edge representing a multiplication by Nj. In this paper, we present a geometric algorithm for producing such trees that is applicable to any integral quadratic form. Although this algorithm does not always yield a tree, we find a few other trees arising from different quadratic forms.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Byungchul Cha, Emily Nguyen, Brandon Tauber,