the sum of ascending powers of a prime can not equal the sum of ascending powers of a different prime.

Question

I was thinking about a question that I can't prove and can't find any proof/counter example for so here it is:

prove the equation: $$ \sum\limits_{i=0}^x p_1^i = \sum\limits_{i=o}^y p_2^i $$ has no solutions where $p_1 $ and $p_2$ are distinct primes and x and y are arbitrary constants.

Answer 1

Note that $1+5+25=1+2+4+8+16$.