How to prove Prime numbers can be expressed as $6k\pm 1$

Question

and

In some of their answer, they mention that some prime $p$ can be expressed as $6k\pm 1$ without proof.

For some few examples i see this, but how i can prove for large $p$?

5, 7, 11, 13, 17, 19, ...

can you give me some proof for this statement?

Please edit the question since 1 is not a prime number, as mentioned by other users. — John W. Smith, Mar 22 '17 at 06:16

score 2 · Accepted Answer · answered Mar 22 '17 at 06:12

All other numbers are either

So if the prime is neither $2$ nor $3$, it cannot belong to any of the categories above. $6k\pm 1$ is what is left.

score 1 · Answer 2 · edited Mar 22 '17 at 06:24

1

$6k$, $6k\pm2$, and $6k\pm3$ are all composite (divisible by 6, 2, and 3, respectively). Thus the primes must be of form $6k\pm1$.

edited Mar 22 '17 at 06:24

Jason Chen

answered Mar 22 '17 at 06:14

Jacob Wakem

1

Please use MathJax when writing mathematics on this site. It looks much better. – Arthur Mar 22 '17 at 06:16
Use the \pm option to do the plus or minus symbol Like this $\pm$. Don't forget the dollar signs as well. – John W. Smith Mar 22 '17 at 06:19
I edited the answer. Waiting for approval. – Jason Chen Mar 22 '17 at 06:19
I cannot approve yet since I don't have enough rep. – John W. Smith Mar 22 '17 at 06:21
I have approved Jason's edit. – Jose Arnaldo Bebita Dris Mar 22 '17 at 06:24

2 Answers2