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Old 2018-06-15, 19:36   #9
Dr Sardonicus
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Feb 2017

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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.
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