Question 64
Prove that there is a constant $c \gt 0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a + i, b + j) \gt 1$ for all $i, j \in \{0, 1, \dots, n\}$, then
$$
\min\{a, b\} \gt c^n \cdot n^{\frac{n}{2}}.
$$
$$
\min\{a, b\} \gt c^n \cdot n^{\frac{n}{2}}.
$$