Prove that there are infinitely many $n \in \mathbb{N}$ such that $6n - 1, 6n + 1$ are composite Let m be a integer. Then if $6n+1$ is a composite number we have that $\operatorname{lcd}(6n+1,m)$ is not just $1$, because then $6n+1$ would be prime. Also this is for $6n-1$. Now I must find all $m$ that this is correct for. I don't know if I'm gonna do it with induction by $n$ or by $m$.
Or is this wrong? Can someone help me please?

Consider the numbers
$$6(35k+1)-1  \quad\text{and}  \quad 6(35k+1)+1$$
for $k=1,2,3,\dots$. The first is divisible by $5$ and the second is divisible by $7$ for all $k$.
