# Prove that the set of all prime numbers of the form $6n+5$ is infinite

Prove that the set of all prime numbers of the form $6n+5$ is infinite.

I'm thinking to show this I would consider $ab\equiv 5 \pmod 6$ so then either $a\equiv 5\pmod 6$ or $b\equiv 5 \pmod 6$. Then try to show any $d=6n+5$ is divisible by a prime of the form $p=6n_0+5$.

• Sounds a good plan. How far have you got? – Lord Shark the Unknown Jan 31 '18 at 4:30
• I see. You have begun deleting your questions that get closed. Always a good option. – Will Jagy Jan 31 '18 at 4:40
• For some finite set of primes that are $\equiv 5 \bmod 6$, create a number that is not divisible by any of them that is also $\equiv 5 \bmod 6$. Then this number is either another prime $\equiv 5 \bmod 6$ or is divisible by such a prime not in the original set. – Joffan Jan 31 '18 at 4:54