# Density of pythagorean triplets

I designed this simple math puzzle last week:

Prove or disprove the following statement:

• Claim: Let $p, q, r, s$ be positive integers such that $p < q$ and $r < s$. Given two rational numbers $\frac{p}{q} < \frac{r}{s}$, there exist positive integers $a, b$ such that $a < b$ and $\frac{p}{q} < \frac{a}{b} < \frac{r}{s}$ and $a^2 + b^2$ is a perfect square.
• If the above statement is true, show an example of $a, b$, expressed in terms of $p, q, r, s$.
• If the above statement is false, construct a counterexample.