TITLE: Find all Pythagorean triples $a<b<c$, where $c=65$.
QUESTION [5 upvotes]: How can one prove that all the Pythagorean triples satisfying this condition have been found? We are working with positive integers a, b, and c.

REPLY [4 votes]: All Pythagorean triples can be found as $$(p^2-q^2)r,2pqr,(p^2+q^2)r$$ Consider different cases for $r=1,5,13$ and for each one consider different pairs of relatively prime $(p,q)$, where $p>q$, $p^2+q^2=65/r$.