Question 63
Consider the infinite, strictly increasing sequence of positive integers $a_{n}$ such that
i. All terms of the sequence are pairwise coprime.
ii. The sum $\frac{1}{\sqrt{a_{1} a_{2}}} + \frac{1}{\sqrt{a_{2} a_{3}}} + \frac{1}{\sqrt{a_{3} a_{4}}} + \cdots$ is unbounded.
Prove that this sequence contains infinitely many primes.
i. All terms of the sequence are pairwise coprime.
ii. The sum $\frac{1}{\sqrt{a_{1} a_{2}}} + \frac{1}{\sqrt{a_{2} a_{3}}} + \frac{1}{\sqrt{a_{3} a_{4}}} + \cdots$ is unbounded.
Prove that this sequence contains infinitely many primes.