So it's a different take on proving there are infinite primes
Given a sequence where any two terms in the sequence are pairwise coprime with each other, how can you prove there are an infinite number of primes combined with the fundamental theorem of arithmetic?
Obviously it follows that if any two of the terms are coprime then their gcd is 1 but I don't see how I can combine this with the fundamental theorem of arithmetic.
Cheers for any help guys
Note: The sequence $\{a_n\}$ defined by $a_n=1$ for all $n$ is a counterexample to this. In what follows, I'll assume that the $a_n$ are $>1$. Easy to generalize to, say $|a_n|>1$ or to something like "infinitely many of the $a_n$ are $>1$ in absolute value.
Let your sequence be $\{a_n\}$ and, for each $n$, define $p_n$ to be the least prime dividing $a_n$.
Your assumption tells us that the $p_n$ are all distinct, since $$p_i=p_j=p\implies p\,|\,\gcd(a_i,a_j)\implies i=j$$
In this way we've produced an infinite sequence of distinct primes.