Elementary Methods in Number Theory Exercise 1.5.9

Show that every prime number except 2 and 3 has a remainder of 1 or 5 when divided by 6.Prove that there are infinitely many prime numbers whose remainder is 5 when divided by 6.

Proof:

(1)Simple.

(2)Suppose there are only finite number of primes whose remainder is 5 when divided by 6,they are
\begin{equation}
p_1,p_2,\cdots,p_n
\end{equation}
It is easy to verify that
\begin{equation}
p_1p_2\cdots p_n\equiv 1\mod 6
\end{equation}
Then let's see
\begin{equation}
p_1p_2\cdots p_n+4
\end{equation}
It is easy to verify that this is a new prime of the form $6k+5$,which leads to absurdity.So there are infinitely many prime of the form $6k+5$