November 13, 2021 by kintali

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”

Chris Grossack | November 16, 2021 at 7:07 AM

My solution was a bit too long for a comment, but you can read it at https://grossack.site/2021/11/16/dense-pythagorean-triples.html ^_^

Reply
- kintali | November 16, 2021 at 4:45 PM
  
  Hi Chris, Your diagrams and the detailed explanations of proofs are awesome 🙂 I have posted a follow up puzzle at https://kintali.wordpress.com/2021/11/16/density-of-pythagorean-triplets-part-2/ Have fun solving it 🙂
  
  Reply
Pingback: Density of pythagorean triplets – Part 2 – My Brain is Open

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