Re: exponential diophantine equations

• To: mathgroup at smc.vnet.net
• Subject: [mg62934] Re: exponential diophantine equations
• From: dh <dh at metrohm.ch>
• Date: Thu, 8 Dec 2005 06:25:38 -0500 (EST)
• References: <dn8qs9\$hj7\$1@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

```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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