From: wolf@doppel.first.gmd.de (Wolfgang Koehler)
Newsgroups: sci.math
Subject: Relations between Quaternions and Number Theory ?
Date: 5 Feb 95 14:18:33 GMT

I have some questions wrt. integer quaternions, ie. quaternions 
 
  q = a + b*i + c*j + d*k   i,j,k units and a,b,c,d integers

Esp. I would like to know if it is possible to give a constructive formula
for the qs. which fulfil the eq.
  -                                 -
  q * q = p, where p is a prime and q is the conjugate of q.

It is well known that there exist exactly 8*(p+1) of them for every prime p,
p != 2.

From a textbook about Gaussian numbers (complex numbers with int. coeff)
I learned that these numbers can be used to construct all integer solutions of 
the eq. :
   a^2 + b^2 = n

It should be straight forward to use quaternions for the extended eq.
   a^2 + b^2 + c^2 + d^2 = n

Has anybody a reference how this can be done ?

Some other related questions:
* Is there an analogy to primes in integers (and Gaussian numbers)
  ie. is it possible to define a set of qs. that are not divisible by others ?

* Is there a analogy to the unique representation  of a number as product of
  primes ?
  Of course, since quaternionic multiplication is not commutative this is
  much more complicated

Any hints or references are welcome !


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Wolfgang Koehler                               wolf@first.gmd.de
GMD-FIRST an der TU Berlin              German National Research Centre 
Tel. (Berlin 030) 6392-1819                 for Computer Science