From: wild_dj@mit.edu (David J Wildstrom) Newsgroups: sci.math Subject: Integer-like Quaternions (was Re: why isn't 1 prime?) Date: 8 Oct 1998 22:02:29 GMT In article , Ken Starks wrote: >I wanted to develop some work I was doing into continued fractions, >and then go on to 'Gaussian Quaternions' or whatever they are called. >I presume they have been invented ??! There are two ways to approach quaternions as an integer-like ring. One is the "Lipschitz" (sp?) or naive domain, consisting of only quaternions with integer components. The other is the "Hurwitz" domain, consisting of both quaternions with integer components _and_ quaternions with exclusively half-integer components. The definitive source on quaternionic number theory is Adolf Hurwitz's _Vorlesungen "Uber die Zahlentheorie der Quaternionen", which has unfortunately never been translated. As a possible alternative, I'd like to suggest my own paper, "Modular Congruence Classes in the Hurwitz Domain", which I could send you in .dvi or .ps format . I encourage you to pursue this much-neglected field. Quoth Hurwitz, concerning the "Zahlentheorie der Quaternionen": I hope these lectures will inspire young colleagues to investigate this area further. (Es wurde mich freuen, wenn diese Vorlesungen etwa jungere Fachgenossen dazu anregen wurden, jenes Gebiet durch weitere Untersuchungen zu fordern) So far very few people have taken him up on his offer. +--First Church of Briantology--Order of the Holy Quaternion--+ | More than just fun-loving surfer guys... watch the VH1 | | special "Endless Harmony" and discover the Beach Boys. | +-------------------------------------------------------------+ | Jake Wildstrom | +-------------------------------------------------------------+