I was just reading a book called Proofs from the Book. It presented the proof given by George Polya to prove that two Fermat primes (numbers of the form $2^{2^n} + 1$) are always relatively prime, which in itself is a very elegant proof. But, to say that it implies there are infinitely many primes does not make sense to me.
15 and 16 are relatively prime but it doesn't imply there are infinitely many primes.
Best Answer
Any sequence $F_n$ of pairwise coprime integers will contain infinitely many primes, because each new $F_n$ has a new prime factor, for all $n\in \mathbb{N}$. In your example, $n=15,16$, but we have infinitely many $n$.