Find three primes numbers Is there any integer $p$ such that $p > 1$ and all three numbers $p$, $p+2$ and $p+4$ are prime numbers? If there are such triples, prove that you have all of them; if there are no such triples, prove why not.
Apart from this, I am given the following information: if $a$ is an integer, then precisely one of the following holds: either $a = 3k$ for some integer $k$, or $a = 3k+1$ for some integer $k$, or $a = 3k+2$ for some integer $k$.
I have observed that, when you sum $p$, $p+2$ and $p+4$, you get $3(p+2)$. But from that, I don't know how to proceed.

If $p$ is not divisible by $3$, then either $p+1$ or $p+2$ must be. (Otherwise, there would be 3 consecutive numbers that are not divisible by $3$, which makes no sense)
If $p+2$ is divisible by $3$, then clearly it is not prime, unless it is $3$. But that would mean that $p=1$, which isn't prime.
But if $p+1$ is divisible by $3$, then $p+4$ must also be, as $p+4=(p+1)+3$. So then $p+4$ is not prime, unless it is $3$. But then $p$ would be $-1$, which is not prime.
So, $p$ must be divisible by $3$, because otherwise, one of the two other numbers is. But if $p$ divides $3$, it must be composite, unless it is $3$.
Therefore, the only three numbers that work are $3,5,7$.
