Can the sums of the first few terms of two geometric progressions with different prime common ratios (start from $1$) be equal?

Question

Consider two geometric progressions with different prime ratios, which both start from $1$. The question is, can the sum of the first $m$ terms of one progression equal to the sum of the first $n$ terms of the other progression?

The sum of the first $n$ terms of a geometric progression starting from $1$ can be written as $\frac{p^{n+1}-1}{p-1}$, when the ratio is $p$. Suppose the two prime numbers are $p$ and $q$. The question can be transformed into whether $$\frac{p^{m+1}-1}{p-1}=\frac{q^{n+1}-1}{q-1}$$ have solutions or not. But I don't know how to deal with it and have no idea about other possible methods to think of the question.

Please give me some ideas.