Prime triples spaced by two

Statement

A prime triple is a triple of prime numbers of the form $p$, $p+2$, $p+4$. For example $3, 5, 7$. Prove that there are no other triples of this form.

Solution

Consider the three numbers $p$, $p+2$, $p+4$ modulo $3$. Their remainders on division by $3$ are, respectively, $p,\quad p+2,\quad p+4 \equiv p,\; p+2,\; p+1 \pmod 3,$ i.e. they cover the three remainders $0, 1, 2$ in some order. In particular one of the three numbers is divisible by $3$.

If that number must be prime and is divisible by $3$, then it must equal $3$. The only way for the one divisible by $3$ to be $3$ is $p = 3$, which gives $3, 5, 7$. For every $p > 3$ one of the three numbers is a multiple of $3$ greater than $3$, hence composite. So $3, 5, 7$ is the only triple. $\blacksquare$