I am looking for infinite set of Diophantine solutions.


1. 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
$$ab-cd=1?$$
2. 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?
3. If not what is the closest permissible?


What is a good bound?



Close integer solutions to $ab-cd=1$?
It is possible with any constant $\lambda>1$ on the place of $\sqrt{2}$. Take $a=t^2$, $d=t^2+t-1$, $b=t^2+2t+1$, $c=t^2+t+1$ for large $t$.


