No answers, please. Hints only.
Given that $n$ is a positive integer and $x=(n+1)!+2$, I want to prove whether or not the sequence $x, x+1, \cdots, x+(n-1)$ contains no primes.
I did a lot of algebra to find that $x+(n-1)$ is no more than just $(n+1)[n!+1]$. But I'm not sure what to do with this, I'm still unsure what this says about the conclusion.
Note that $(n+1)!$ is the product of the first $n+1$ positive integers. Can you show that $2$ divides $(n+1)!+2$? Can you generalize from here?