Friday, October 12, 2012: 4:40 AM
Hall 4E/F (WSCC)
Solutions to x2 + y2 = z2 are widely explored in the integers; however, much is left to
be discovered in other rings, such as the Gaussian integers and the real Quaternions. Our
research focuses on the Quaternions and examines conditions under which we can find families
of Pythagorean triples in that ring. Our method consists of taking a Pythagorean triple (a, b, c) in
HZ and constructing a t in HZ dependent on a, b, and c such that (a-t, b-t, c+t) is a Quaternion
Pythagorean triple. To find necessary and suficient conditions on t, we utilize the results that for
b in Im(HZ), b2 is an integer and b2 = -N(b), where Im(HZ) denotes a pure imaginary Quaternion.
be discovered in other rings, such as the Gaussian integers and the real Quaternions. Our
research focuses on the Quaternions and examines conditions under which we can find families
of Pythagorean triples in that ring. Our method consists of taking a Pythagorean triple (a, b, c) in
HZ and constructing a t in HZ dependent on a, b, and c such that (a-t, b-t, c+t) is a Quaternion
Pythagorean triple. To find necessary and suficient conditions on t, we utilize the results that for
b in Im(HZ), b2 is an integer and b2 = -N(b), where Im(HZ) denotes a pure imaginary Quaternion.