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exponential diophantine equations - correction!


Correction. In the example below, I had typed +7.  It should have been -7.

I'm trying to find out how to use Mathematica to find solutions to
exponential diophantine equations like the following:

          (5x)^2 - 2^k * 3 * (5+k)^2 - 131 * k - 7 = 0.   I want to obtain
solutions for x and k. (One solution is x = 31, k = 6, but I didn't find
this using Mathematica!)

I'm not very skilled in Mathematica. I tried Solve and Reduce but I just
get either

The equations appear to involve the variables to be solved for in an
essentially non-algebraic way.
or
This system cannot be solved with the methods available to Reduce.

I'm trying to find a way to get at least one or two solutions. Is there a
way in Mathematica to do this? I know that exponential ones are not easy...

Thanks for any help.

Andrea



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