Are all the primes greater than $3$ in form of $6n\pm1$

Question

today I've encountered the following excercise:

Show that all the primes greater than $3$ are in form of $6n\pm1$...

My initial thought was that numbers are in form of $6n,\quad 6n\pm1,\quad 6n\pm2, \quad 6n\pm3$ in respect to $6$ and $6n\pm2=2(3n\pm1),\quad 6n\pm3=3(2n\pm1)$ so since they are multiples of $2$ and $3$ they can't represent primes. This however didn't completely satisfy me, because I've shown that other forms in respect to $6$ aren't primes, but that doesn't necessarily mean that $6n\pm1$ must be prime. I thought perhaps I can do induction but I don't know what inital property to use for a start, or perhaps I am overseeing it right now.

Thanks in advance!

Answer 1

Are all primes greater than $3$ of the form $6n\pm 1$?

Yes, because the only other forms are $6n$, $6n\pm 2 = 3(2n\pm 1)$ and $6n + 3 = 3(2n+1)$ which can't be primes.

Are all integers of the form $6n\pm 1$ prime? Of course not. $25 = 4*6+1; 35=6*6-1;49=6*8+1$ etc. show they are not.

Were you asked to prove that all primes greater than $3$ are of the form $6n\pm 1$? Yes, you were. Did you? Yes, you did.

Were you asked to prove that everything of the form $6n \pm 1$ was a prime? No, you were not. Were you able to? No, you couldn't. Can anyone? No, because it isn't true. Were you expected to? No, you were not because it isn't true.

Answer 2

It's because $6n+3$ is divisible by $3$

(and obviously, $6n$, $6n+2$, and $6n+4$ are all divisible by $2$)

So any prime must be of the form $6n+1$ or $6n+5$ (i.e. $6n-1$) ... which of course does not mean that all such numbers are primes .. but that wasn't the question.