TITLE: Is it true that $(p,q)=(q',p')$?
QUESTION [5 upvotes]: If $a,b,p,q,p',q'\in \mathbb{Z}$ satisfy $a(p^2+q^2)+bp'q'=a(p'^2+q'^2)+bpq$, is it true that $(p,q)=(q',p')$?
EDIT: This is wrong, I have made a new topic here: Which linear transformations preserve this?

REPLY [2 votes]: The equation you want to solve is equivalent to
$$ a(p^2+q^2-p^2-q'^2) = b(pq-p'q') $$
Unless I'm missing something, you can choose $p,q,p',q'$ essentially arbitrarily and then set
$$ a=pq-p'q' \qquad b=p^2+q^2-p'^2-q'^2 $$