Prove that the only prime triple is 3, 5, 7 [duplicate]
Every non-negative integer $n$ can be written in the form $n=6x+k$, where $x$ is a non-negative integer and $k\in\{0,1,2,3,4,5\}$
$n=6x,6x+2,$ or $6x+4 \implies 2\mid n$, while $n=6x$ or $6x+3 \implies 3\mid n$, which leaves only $n=6x+1$ and $n=6x+5=6(x+1)-1$ for integers greater than $3$ and not divisible by $2$ or $3$, and as a special case for integers greater than $3$ and not divisible by any smaller prime i.e. primes,
so for $p\in \Bbb P$ such that $p>3$,
$p+2 \in \Bbb P \implies p=6x+5 \implies p+4=6(x+1)+3 \implies 3\mid p+4$.