I'm trying to prove that a number with two distinct prime factors can't be friends with another number with the same prime factors.
One way I could prove this is that there'd be only one example where $$\sum_{i=0}^np^i=\sum_{j=0}^mq^j$$
That example, preferrably, would be $2^0+2^1+2^2+2^3+2^4=5^0+5^1+5^2=31$, which fails to fit other conditions necessary to construct that pair of friends.
Through a little bit of computing power, I was unable to find examples for $p<300,n<10$, which leads me to believe it may be the only example. However, I'm completely lost on a continuation, if there is one, and whether this is just a case of the XY problem, and I should drop this line of reasoning and move elsewhere.
Nope!
$$\sum_{i=0}^2{90^i}=1+90+90^2=8191=1+2+2^2+2^3+\dots +2^{12}=\sum_{j=0}^{12}{2^j}$$
As is mentioned in the comments this question is duplicate.