I am looking for infinite set of Diophantine solutions.
Suppose we require $$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)\leq\sqrt 2\min(a,d)$$ $$a,b,c,d\in\mathbb Z$$ then can we still find solutions to $$ab-cd=1?$$
Is it possible to do this if only $$a,b,c,d\in\mathbb Z$$ $$0<\min(a,d)<\max(a,d)<\min(b,c)<\max(b,c)$$ $$ad<ab+cd\leq2ad\leq2bc$$ holds?
If not what is the closest permissible?
What is a good bound?