The pythagorean theorem in quaternion and octonion spaces

The Pythagorean theorem x²+y²=z² is usually stated over the integers Z , a subring of the reals R , and it holds true for infinitely many solutions. We explore the theorem over subrings of other number spaces, such as H (quaternions) and O (octonions). We present several results mainly for L (Lipschitz quaternions) and provide a geometric interpretation of the theorem in the subring L . Some results for the corresponding subring G of O are also presented. Finally, we also present some results for the rings H/Zₚ and O/Zₚ .