Two sets of positive integers $A, B$ are called connected if they are not empty and for all $a \in A, b \in B$, number $ab + 1$ is a perfect square. i) Given $A =\{1, 2,3, 4\}$. Prove that there does not exist any set $B$ such that $A, B$ are connected. ii) Suppose that $A, B$ are connected with $|A|,|B| \ge 2$. For any $a_1 > a_2 \in A$ and $b_1 > b_2 \in B$, prove that $a_1b_1 > 13a_2b_2$.