Pythagorean Triples #
The main result is the classification of Pythagorean triples. The final result is for general
Pythagorean triples. It follows from the more interesting relatively prime case. We use the
"rational parametrization of the circle" method for the proof. The parametrization maps the point
(x / z, y / z) to the slope of the line through
(-1 , 0) and
(x / z, y / z). This quickly
(x / z, y / z) = (2 * m * n / (m ^ 2 + n ^ 2), (m ^ 2 - n ^ 2) / (m ^ 2 + n ^ 2)) where
m / n is the slope. In order to identify numerators and denominators we now need results showing
that these are coprime. This is easy except for the prime 2. In order to deal with that we have to
analyze the parity of
n and eliminate all the impossible cases. This takes up
the bulk of the proof below.
z form a Pythagorean triple if
x * x + y * y = z * z.
Pythagorean triples are interchangable, i.e
x * x + y * y = y * y + x * x = z * z.
This comes from additive commutativity.
The zeroth Pythagorean triple is all zeros.
A triple is still a triple if you multiply
by a constant
(k*x, k*y, k*z) is a Pythagorean triple if and only if
(x, y, z) is also a triple.
A Pythagorean triple
x, y, z is “classified” if there exist integers
k, m, n such that
x = k * (m ^ 2 - n ^ 2)and
y = k * (2 * m * n), or
x = k * (2 * m * n)and
y = k * (m ^ 2 - n ^ 2).
A primitive pythogorean triple
x, y, z is a pythagorean triple with
Such a triple is “primitively classified” if there exist coprime integers
m, n such that either
x = m ^ 2 - n ^ 2and
y = 2 * m * n, or
x = 2 * m * nand
y = m ^ 2 - n ^ 2.
A parametrization of the unit circle #
For the classification of pythogorean triples, we will use a parametrization of the unit circle.
A parameterization of the unit circle that is useful for classifying Pythagorean triples.
(To be applied in the case where
K = ℚ.)
x is odd and
z is positive we get a slightly more precise classification of
the pythagorean triple
x ^ 2 + y ^ 2 = z ^ 2