Solutions of Pythagorean Equation/General

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Theorem

Let $x, y, z$ be a solution to the Pythagorean equation.

Then $x = k x', y = k y', z = k z'$, where:

$\left({x', y', z'}\right)$ is a primitive Pythagorean triple
$k \in \Z: k \ge 1$


Proof

Let $\left({x, y, z}\right)$ be non-primitive solution to the Pythagorean equation.


Let:

$\exists k \in \Z: k \ge 2, k \mathrel \backslash x, k \mathrel \backslash y$

such that $x \perp y$.

Then we can express $x$ and $y$ as $x = k x', y = k y'$.

Thus:

$z^2 = k^2 x'^2 + k^2 y'^2 = k^2 z'^2$

for some $z' \in \Z$.


Let:

$\exists k \in \Z: k \ge 2, k \mathrel \backslash x, k \mathrel \backslash z$

such that $x \perp z$

Then we can express $x$ and $z$ as $x = k x', z = k z'$.

Thus:

$y^2 = k^2 z'^2 - k^2 x'^2 = k^2 y'^2$

for some $y' \in \Z$.


Similarly for any common divisor of $y$ and $z$.


Thus any common divisor of any pair of $x, y, z$ has to be a common divisor of Integers of the other.

Hence any non-primitive solution to the Pythagorean equation is a constant multiple of some primitive solution.

$\blacksquare$


Historical Note

It is clear from the cuneiform tablet Plimpton $\mathit { 322 }$ that the ancient Babylonians of $2000$ BCE were familiar with this result.

The complete solution of the Pythagorean equation was known to Diophantus of Alexandria.

It forms problem $8$ of the second book of his Arithmetica.

It was in the margin of his copy of Bachet's translation of this where Pierre de Fermat made his famous marginal note that led to the hunt for the proof of Fermat's Last Theorem.


Sources