Let $S_1=\sum_{i=0}^{n} p^i = \frac{p^{n+1}-1}{p-1}$ and $S_2=\sum_{i=0}^{m} q^i = \frac{q^{n+1}-1}{q-1}$ be two sums of geometric series, and $\gcd\left(S_1,S_2\right)$ its greatest common divisor. What is it known about the non-trivial upper bounds of $\gcd\left(S_1,S_2\right)$?
The question is related to the sum of divisors function, as $\sigma_1(p^n)=S_1$ and $\sigma_1(q^m)=S_2$ when $p,q$ are prime numbers.
I am particularly interested on a non-trivial upper bound for the case $p,q$ being both odd prime numbers and $n,m$ being both even, but if there are some other results or generalizations I would be interested too.
I appreciate both references and concrete bounds proofs.
Thanks in advance!