Prove that there is a constant $c > 0$ with the following property: If $a, b, n$ are positive integers such that $ \gcd(a+i, b+j)>1 $ for all $ i, j\in\{0, 1,\ldots, n\} $ then
$$ \min\{a, b\}>c^n\cdot n^{\frac{n}{2}}$$
I've reduced this to the following problem:
$$\sum_{k = 0}^{\infty} \frac{1}{p_k^2} \leq \frac{1}{2} $$ where $p_k$ is the $k$th prime. How do I proceed from here?
(Numerical estimation says that the sum is about 0.45, so it would be quite a tight bound).
Here is one approach:
Note that $$\sum_{k=1}^{\infty} \frac{1}{p_k^2} \leq \sum_{i = 1}^{k-1} \frac{1}{p_i^2} + \int_{p_k -1}^{\infty} \frac{1}{x^2} = \sum_{i=1}^{k-1} \frac{1}{p_i^2} + \frac{1}{p_k - 1}.$$
Since $1/(p_k - 1)\to 0$ as $k \to \infty$, and since the sum is strictly $< 1/2$, we see that taking $k$ large enough we will be able to make the RHS less than $1/2$. In fact, taking $p_k = 17$ seems to do the job (if I didn't miscalculate).