Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
Is there a way to give a sharp estimate?
Given $m,n\in\Bbb N$ with $m<n$, how many pythagorean triples $p^2+r^2=q^2$ satisfy $$m\leq p<r\leq n?$$
Is there a way to give a sharp estimate?
See my preprint. Page 12 (this has since been published, but preprint is easiest to access).
For each type of Pythagorean Triples $p=2xy$, $r=x^2-y^2$ and $p=x^2-y^2$, $r=2xy$ your inequalities describe a simple region on $Oxy$ plane. Boundary consits from the straight line $2xy=x^2-y^2$ and hyperbolas $2xy=p, x^2-y^2=r$ and $x^2-y^2=p, 2xy=r$.