# Math Help - Euclid's Theorem

1. ## Euclid's Theorem

How do you prove that there exists infinitely many primes p such that p = 2 mod 3 ?

I know that we can use the Euclid's Theorem but I have no idea how to do this kind of question, can anyone give me some hints?

Thanks alot

2. Originally Posted by knguyen2005
How do you prove that there exists infinitely many primes p such that p = 2 mod 3 ?
Assume there are finitely many $p_1,...,p_n$.
Form $N = 3p_1....p_n - 1$.

We know that $N$ has an odd prime divisor $p$. If all its prime divisiors were of the form $3k+1$ then $N$ would be of the form $3k+1$ and impossibility. Therefore there is a prime divisor $p$ of the form $3k+2$. Thus, $p|N$ and it must be amongst $p_1,...,p_n$. But that leads to $p|1$ which is a contradiction.

3. thank you very much ThePerfectHacker