Intro:
Above is a unit circle, .
- First (trivial) solution is .
- For second (non-trivial) solution, QR has a rational slope and we plug this .
This gives (non-trivial) solution of .
Simple enough. Now the exercise.
I'm having problem with a minus in part:The parameter in the pair runs through all rational numbers if and run through all pairs of integers.
Deduce that if is any Pythagorean triple then
for some integers and .
How would I use the previous exercise to prove that Euclid's generates all triples?Use previous exercise to prove Euclid's formula for Pythagorean triples.