By knowing the number in a set are all coprimes, is it possible to prove the inifinity of primes? If I know all the numbers in a set are coprime, for example, the set of Fermat numbers, is it possible to prove the infinity of the primes? 
Thanks. 
 A: If the set is infinite, then yes. The reason is simple : by the uniqueness of factorization of integers into primes, each member of your set has a prime factor that no other members of your set has. Therefore there must be at least as many primes as the size of your set. If your set is infinite, you are done.
Hope that helps,
A: If $S$ is an infinite set of integers ( such that any two distinct members of it are relatively prime, then the set of prime numbers must be infinite. Given such a set $S$, here is a procedure to produce an infinite list of distinct primes. Since $S$ is infinite, choose $a_1\in S$ different than $1,0,-1$. Let $p_1$ be a prime that divides $a_1$. Now suppose you had already chosen $a_1,\cdots ,a_n\in S$ and primes $p_1,\cdots, p_n$ such that $p_k$ divides $a_k$ for all $k=1,\cdots, n$, and the primes are distinct. Let $a_{n+1}\in S-\{a_1,\cdots ,a_n\}$ and different than $1,-1,0$. Let $p_{n+1}$ be any prime dividing $a_{n+1}$. Since $gcd(a_k,a_{n+1})=1$ for all $k=1,\cdots ,n$, it follows that $p_{n+1}\notin \{p_1,\cdots ,p_n\}$. This process never terminates, thus producing an infinite list of distinct primes.
