Why are primes of the form $6k+1, 6k-1$ where the prime is $\geq 3$

Question

I recently came to know that primes are of form $6k+1,6k-1$ for primes greater than three. Why is this so? I tried my hand on it could not really understand about it. I have also heard of Dirichlet's theorem but can there be any elementary such way to show this?

Possible duplicate of Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$ — Dietrich Burde, Apr 28 '19 at 15:42

score 1 · Accepted Answer · answered Apr 28 '19 at 15:38

1

What are the other possibilities? Numbers of the form $6k$, $6k+2$, or $6k+4$ are all divisible by $2$, while numbers of the form $6k+3$ are divisible by $3$ and thus are not prime if $k>0$.

answered Apr 28 '19 at 15:38

rogerl

22,399

Oh yeah i see thank you! – Apr 28 '19 at 15:39

Matt Samuel · Answer 2 · 2019-04-28T15:58:30.520

0

$6k,6k+2,6k+3,6k+4$ cannot be prime for $k>0$ because they are divisible by $6,2,3,2$ respectively. Thus only $6k+1$ and $6k+5$ are left.

edited Apr 28 '19 at 15:58

answered Apr 28 '19 at 15:39

Matt Samuel

58,164

They cannot be prime for $\mathbf {k>0}$ – J. W. Tanner Apr 28 '19 at 15:57

Why are primes of the form $6k+1, 6k-1$ where the prime is $\geq 3$

2 Answers2