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.

Leave your solutions in the comments. Have fun solving it.

3 thoughts on “Density of pythagorean triplets

  1. Pingback: Density of pythagorean triplets – Part 2 – My Brain is Open

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