About two exponential diophantine equations....

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g.pinedav@gmail.com - 02 Apr 2009 04:42 GMT

Hi There,

I have the following question: I believe that for any natural D\ge3

\frac{2\times3^{D}(2\times3^{D}-3)}{2D+1} and \frac{(2\times3^D-3)}{D}
are not integer expressions simultaneously.

How can I prove it? I have tried to prove it with no luck so far. I am
not aware of any theory about this type of problem.

I would appreciate any help in this regard,
Guillermo

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Kevin Buzzard - 02 Apr 2009 18:53 GMT

> Hi There,
>
> I have the following question: I believe that for any natural D\ge3
>
> \frac{2\times3^{D}(2\times3^{D}-3)}{2D+1} and \frac{(2\times3^D-3)}{D}
> are not integer expressions simultaneously.

Nothing deep here, just some preliminary remarks.

So for each of these fractions, computer experiments seem to indicate
that there's a sparse set of D for which the fraction is an integer.
If D=(3^t-1)/2 for t>=1 and if I've read the TeX correctly then the
first one is an integer, but there are more solutions than that (e.g. D=22).
There seem to be an even sparser set of
solutions to the second equation [I found
1
3
51
329
172299
1723443
]

so your belief may well be true.
But it might be a hard question! There are plenty of "elementary"
questions like this which turn out to be hard. What is the source
of the question? If it's e.g. an old Olympiad problem then there will
surely be an elementary solution. But if you've just "made it up"
in some way then it *might* be "harder than Fermat's Last Theorem".
Or there might be a slick 3-line solution. I'm reluctant to spend
too much time on the problem though if there is no guarantee that
it is solvable using current techniques :-)

Kevin

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Math Forum / Mathematics / Research / April 2009

About two exponential diophantine equations....