Solutions of Pythagorean Equation/General

Theorem

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

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

$k \in \Z: k \ge 1$

Let $\tuple {x, y, z}$ be non-primitive solution to the Pythagorean equation.

Let:

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

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:

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

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.

$\blacksquare$

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

The proof was provided by Euclid and 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.

1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {A}.9$: Hypatia (A.D. $\text {370?}$ – $\text {415}$)