# Why can't FindInstance find a solution to this equation?

FindInstance[680 == 1/3 2^m (-1 + 4^v), {m, v}, Integers]


FindInstance seems to be unable to find the integer solution ${m = 3,v = 4}$ in the equation above (failing with the message:

FindInstance::nsmet : The methods available to FindInstance are insufficient to find
the requested instances or prove they do not exist.


Is there are technique that can be used to find integer solutions to equations like the above?

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FindInstance[680 == 1/3 2^m (-1 + 4^v) && 0 < m < 10, {m, v}, Integers]

I get as output {{m -> 3, v -> 4}}. So, by restricting the equation a bit, you can find an (integer) solution to such equations. But you should have a feeling for the solution (if you have bad luck and use only 4 < m < 10 then you will not find the solution).
A systematic search returns only one solution {m->3, v->4}.