View Single Post
Old 2018-06-15, 19:36   #9
Dr Sardonicus
 
Dr Sardonicus's Avatar
 
Feb 2017
Nowhere

137916 Posts
Default Algebraic formulas are algebraic formulas...

Substituting Gaussian integers z1 and z2 into the usual parametric formulas for Pythagorean triples,

(A, B, C) = (z12 - z22, 2*z1*z2, z12 + z22)

We assume that z1 and z2 are nonzero. We obtain primitive triples if gcd(z1, z2) = 1 and gcd(z1 + z2, 2) = 1. The latter condition rules out z1 and z2 being complex-conjugate.

We obviously obtain thinly disguised versions of rational-integer triples when one of z1 and z2 is real, and the other is pure imaginary.

Obviously A, B, and C are real when z1 and z2 are rational integers.

Clearly B is real when z2 is a real multiple of conj(z1).

Also, B/C is real when z2/z1 is real, or |z1| = |z2|.

A/C is only real when z2/z1 is real.

The nontrivial primitive solutions with A, B, C all complex having the smallest coefficients appear to be

z1 = 1, z2 = 1 + I: A = 1 - 2*I, B = 2 + 2*I, C = 1 + 2*I

and variants.
Dr Sardonicus is online now   Reply With Quote