Square of Pythagorean Prime is Hypotenuse of Pythagorean Triangle
Theorem
Let $p$ be a Pythagorean prime.
Then $p^2$ is the hypotenuse of a Pythagorean triangle.
Proof
By Fermat's Two Squares Theorem, a Pythagorean prime, $p$ can be expressed in the form:
- $p = m^2 + n^2$
where $m$ and $n$ are (strictly) positive integers.
- $(1): \quad m \ne n$, otherwise $p$ would be of the form $2 m^2$ and so even and therefore not a prime.
- $(2): \quad m \perp n$, otherwise $\exists c \in \Z: p = c^2 \paren {p^2 + q^2}$ for some $p, q \in \Z$ and therefore not a prime.
- $(3): \quad m$ and $n$ are of opposite parity, otherwise $p = m^2 + n^2$ is even and therefore not a prime.
Without loss of generality, let $m > n$.
From Solutions of Pythagorean Equation, the triple $\tuple {2 m n, m^2 - n^2, m^2 + n^2}$ forms a primitive Pythagorean triple such that $m^2 + n^2$ forms the hypotenuse of a primitive Pythagorean triangle.
Thus we have that $p^2 = r^2 + s^2$ for some (strictly) positive integers $r$ and $s$.
Similarly, we have that $r \ne s$, otherwise $p^2$ would be of the form $2 r^2$ and so not a square number.
Without loss of generality, let $r > s$.
From Solutions of Pythagorean Equation, the triple $\tuple {2 r s, r^2 - s^2, r^2 + s^2}$ forms a Pythagorean triple such that $p^2 = r^2 + s^2$ forms the hypotenuse of a Pythagorean triangle.
$\blacksquare$
Sources
- 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $13$
- 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $13$