TITLE: $n=pq$ with $p<q$ odd primes is there a fast way to determine if $p-1|q-1$ without knowing $p$ and $q$
QUESTION [17 upvotes]: Let $n=pq$ with $p,q$ odd primes $p<q$
If we only know $n$ but not $p$ and $q$, is there a fast way to determine if $p-1|q-1$ without factorizing $n$ ?

REPLY [14 votes]: Let $n = pq$
if  $p-1 \mid q-1$
1:  $p-1 \mid n-1$
2:  $q-1 \not\mid n-1$ (if this false, $n$ is a Carmichael number with 2 divisors) 
3: $x^{n-1} = 1 \pmod p$ for any $x$ where $x$ is coprime with $n$ $\implies$ $x^n = x \pmod p$
So, we can do this steps: 
Get random $x$ from $[0, n-1]$
Calculate $t = (x^n-x) \bmod n$.  It is not $0$ with the probability about $\frac{\phi(q-1)}{q-1}$
$t=0 \pmod p$. So, greatest common divisor of $n$ and $t$ is $p$ if  $p-1 \mid q-1$
So, if $\gcd \{\,n,t\,\}$ is $1$ - we can say "no".
if $\gcd \{\,n, t\,\} > 1$, we get the factorization and can check $p-1 \mid q-1$ manually