Proving that there are infinitely many primes is fairly simple:
- Assume that there is a finite number of primes.
- Let $G$ be the set of all primes $P_1,P_2,\ldots,P_n$.
- Compute $K = P_1 \times P_2 \times \cdots \times P_n + 1$.
- If $K$ is prime, then it is obviously not in $G$.
- Otherwise, none of its prime factors are in $G$.
- Conclusion: $G$ is not the set of all primes.
I thought I could use a similar method in order to prove:
- There are infinitely many primes $P_i\equiv1\pmod6$
- There are infinitely many primes $P_i\equiv5\pmod6$
But it doesn't appear to be that simple... any ideas?