Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.
We proceed inductively by constructing a sequence $\{n_k\}$ of numbers that work. Notice that for the base case, $2^2-3$ and $2^3-3$, $2^4-3$ are relatively prime.
Now, let $S$ be the set of all $p_i$ such that $p_i \mid 2^{n_k} - 3$ for some $k$. Then we can pick $n_{k+1} = \prod_{p_i \in S} (p_i - 1)$. This works because $2^{n_{k+1}} - 3 \equiv -2 \pmod {p_i}$ for any $p_i \mid n_k$, and thus is relatively prime to all previous elements.