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This is extension of my previous question How to prove Prime numbers can be expressed as $6k\pm 1$

The main answer from the post is list all $6k\pm1, 6k \pm2, 6k +3, $ and see wheter it can be decompose into other numbers, $i.e$, $2$ or $3$.

At this stage, i have a question why $6$?

How about 4, 5, 7, or any other numbers and forms like $ak+b$?

Is there any reason for $6$?

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  • If you understand the logic behind the answers to the other question, you may apply it to 4, 5, 7, and see what happens. – Ivan Neretin Mar 22 '17 at 07:42
  • You can continue to do this indefinitely. All primes (except for 2 and 3) are of the fork $36k \pm i$ where $i$ is $1,5,7,11,13,17$. That's just not as pithy. – fleablood Mar 22 '17 at 18:29

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To have $ak \pm m $ describe all primes, then $m $ and $a-m $ must be the only two numbers smaller than $a$ that are relatively prime to $a $.

For $2$, $2k+1$ covers all primes (except 2) because the only other cases are $2k $ and those are composite (except 2).

Same with $3k\pm 1$. The only other case is $3k $.

$4k\pm 1$ describes all primes (except 2) because the only other case is $4k$ and $4k +2$. But $2$ is not relatively prime to $4$ so all $4k+2$ will be composite.

We can't do anything with $5k \pm i $. $5k$ must be composite, but as $1,2,3,4$ are all relatively prime to $5$ we can't say any of the $5k+i $ must be composite. $5k+i $ could be prime for all $1,2,3,4$.

$6k \pm 1$ describe all primes because none of $i=0,2,3,4$ are relatively prime to $6$ so all $6k +i $ are composite.

We can't do anything with $7k \pm m $ because all $i=1,2,3,4,5,6$ are realitively prime so none of $7k+i $ must be composite.

For $8$ $8k\pm 1$ and $8k\pm 3$ describe all primes (except 2) because all other $i=0,2,4,6$ are not relatively prime to $8$ so $8k+i $ must be composite. But $\pm 1$ and $\pm 3$ are both relatively prime so we need both of them and we can't make this a single expression.

And for no other number higher than $6$, can we have only one pair of relatively prime numbers. So $6k\pm1$ is the last such single expression that describes all primes.

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