Can we say that if $p$ and $q$ are distinct prime number diving $n$
$\Omega(gcd(p-1,q-1)) \leq \Omega(n)$
Where $\Omega(n)$ denotes the number of prime powers dividing $n$
closed as too localized by Gjergji Zaimi, Franz Lemmermeyer, Bill Johnson, S. Carnahan♦ Mar 18 '11 at 16:47
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No: take $n=211*2311$, $p=211, q=2311$. In fact there is no bound of the L.H.S. in terms of the R.H.S. in your inequality. Take any sequence of primes $p_1,...,p_s$. Let $a=p_1\cdot...\dot p_s$. By Dirichlet theorem there are two primes $p=ak+1, q=am+1$, $k < m$. Let $n=pq$. Then $\Omega(n)=2$ while $GCD(p-1,q-1)$ is divisible by $p_1,...,p_n$, so $\Omega(GCD)\ge n$.