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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.

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Based on the answer, I think you should qualify the statement with $p_1$ and $p_2$ both greater than 2. – tpb261 May 26 '14 at 7:35
okay, anybody got any suggestions for primes greater than 2? – maxG795 May 26 '14 at 15:39
up vote 18 down vote accepted

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

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Seems to be the only solution up to $10^6$. – lhf May 26 '14 at 2:45

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