Simple question

[Lior]

Please prove that the equation m^3 - n^2 - 2 =0 has only one solution.
(and the only solution is m=3 n=5).

Thanks.

P.S: m and n are natural numbers.

[Guv]

This is likely to be very tuff to prove. Do you have a good reason to believe that it is true?

I did a little numerical experimentation and found some values of m for which n was close to an integer.

One and only one integer solution to such an equation seems a bit strange. If one solution, why not more?

Such problems are known as Diophantine equations or diophantine analysis. If you try a search for Diophantine, you might find some methods of attacking such problems. I have seen a site that solves linear Diophantine equations and gives some clues on how it is done.

[Lior]

Well...I've came up with this equation after reading "Fermat's Last Theorem - By Simon Sing" (Highly recommended).

It was written there, that Fermat proved that the number 26 is the only natural number located between two powers.
Because:
3^3=27
5^2=25

Now, let X be the number (which Fermat proved X could be only 26).

Let M^3=X+1 and N^2=X-1 and you'll get the equation:

M^3 - N^2 = X+1 - (X-1)

M^3 - N^2 = 2

M^3 - N^2 - 2 = 0.

Now we have to prove that (m=3 and n=5) is the only integer solution.

[Guv]

From what you have said, it seems to me that the Fermat theorem is a proof that the following equation has only one solution for which m & n are integers.

m^3 - n^2 = 2

That seems to be a remarkable theorem. Lots of luck understanding the proof Fermat devised. He was incredible. It is possible that he had methods and/or knowledge that has never been rediscovered.

Are you sure he proved the 25, 26, 27 theorem about 26 being the only integer between two powers?

[Lior]

I am sure he PROVED the 26 thingy.
(Unless the author wrote a lie).

btw, the author also mentioned the proof is quite complicated, that's what challenged me a bit, and I came up with the equation I mentioned. but now I see, I cannot prove the equation I came to. I thought that maybe you could gimmi a direction or something.

[DaoK]

I haev the prove in my Mathematic Book but this book it's at school, tommorow I will give your the answer.

[Guv]

Did you expect the proof to be simple? Fermat did some esoteric work which is incomprehensible to ordinary mortals.

I repeat that the Fermat proof is a proof that your equation has no other integer solutions.

I do not beleive that the author of the book published a lie. It is just that some of the subject matter of number theory is so difficult that it can be misinterpreted.

[NotLKH]

Heh, I found your proof, but it acknowledges itself as potentially incomplete.

BTW, it uses the form y^3 = x^2 + 2, instead of m^3 - n^2 - 2 = 0.


-Lou

[[praetorian]]

Doesn't seem to hard........=)