Divisibility by primes Suppose that $n$ is a natural number, $n \ge 2$, and $n$ satisfies:


*

*For each prime divisor $p$ of $n$, $p^2$ does not divide n.

*If $p$ is prime, $p$ divides $n$ if and only if $(p − 1)$ divides $n$.


Compute $n$.

$p-1$, $p$ are both primes if that is $2$ and $3$
so
$p_0=2$
$p_1=p_0+1=3$
$p_2=p_0p_1+1=7$
$p_3=p_0p_1p_2+1=43$
$p_4=1807$
I am kind of lost what to do next.
Thank you very much

Hint:  In addition to $2\cdot3\cdot7\cdot43+1=1807$, you need to consider whether $2\cdot43+1$, $2\cdot3\cdot43+1$ and $2\cdot7\cdot43+1$ are prime as well.
