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

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- Oct 2nd 2008, 06:47 AMknguyen2005Euclid'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 - Oct 2nd 2008, 07:58 AMThePerfectHacker
Assume there are finitely many $\displaystyle p_1,...,p_n$.

Form $\displaystyle N = 3p_1....p_n - 1$.

We know that $\displaystyle N$ has an odd prime divisor $\displaystyle p$. If all its prime divisiors were of the form $\displaystyle 3k+1$ then $\displaystyle N$ would be of the form $\displaystyle 3k+1$ and impossibility. Therefore there is a prime divisor $\displaystyle p$ of the form $\displaystyle 3k+2$. Thus, $\displaystyle p|N$ and it must be amongst $\displaystyle p_1,...,p_n$. But that leads to $\displaystyle p|1$ which is a contradiction. - Oct 2nd 2008, 11:52 AMknguyen2005
thank you very much ThePerfectHacker