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Re: exponential diophantine equations


Hi Andrea
see below,
Daniel

Andrea wrote:
> 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!)

You did not find it because it is not a solution:
(5x)^2 - 2^k*3*(5 + k)^2 - 131*k + 7 /. {x -> 31, k -> 6}
gives 14 and not zero.

There is no solution as you may convince yourself:
(5x)^2 == -(- 2^k*3*(5 + k)^2 - 131*k + 7)
the left side is always positive, therefore, we only need to consider k 
for which the right side is positive, that is k>=0.
In this region the left side is always smaller than the right side that 
grows exponentially.

Daniel

> 
> 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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