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Fermat claimed that $x ^ 3-y ^ 2 = 2$ only has one solution $(3,5)$, but did not write a proof.
Who can provide a proof that a high school student can accept?


Thank you for your help An answer given by the Chinese friends: similar to the integer division algorithm, but the Chinese, in front of first give some basic properties of the final is proved.Please look at. enter image description here

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Well, it depends on the high school student :) –  user17762 May 24 '12 at 7:17
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My proof won't fit in the margin. –  copper.hat May 24 '12 at 7:18
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I was able to check that $(3,5)$ is indeed a solution. –  Fabian May 24 '12 at 7:26
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You should be careful to mention you want integer solutions. –  Steven-Owen May 24 '12 at 7:33
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A solution using the fact that $\mathbb Z[\sqrt{-2}]$ is an UFD is given in the book Titu Andreescu,Dorin Andrica,Ion Cucurezeanu: An Introduction To Diophantine Equations, p.169. However, this is not accessible to high school students. –  Martin Sleziak May 24 '12 at 7:38

2 Answers 2

The following paper comes as close as I could find to be self-contained and ""basic"" in its proof. Please do note they prove there that $\,(5,3)\,$ is the only integer solution of the diophantine eq. $\,y^3-x^2=2\,$ , and that they use the notation $\,x\wedge y$ to denote the gcd of two integers $\,x\,,\,y\,$

Added: Oops, sorry! Didn't notice I didn't write down the link. Here it is http://www.normalesup.org/~baglio/maths/26number.pdf

Please notice the paper seems to be written by advanced H.S. students and/or beginning university ones, and the language is rather sloppy.

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We're waiting... –  TonyK May 24 '12 at 11:51
    
Waiting I would like to Fermat Is deceiving you what? Solve elementary mathematics are very interested in mathematics master, I know there are Euler website eulersociety.org There are other master site? –  tianzhidaosunyouyu May 25 '12 at 6:07
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But, as far as I perceive, this still avails of the unique factorising property to resolve the problem, while showing no attempt to evident this assumption. Well, t'is what being elementary means? Assume something without proof? Might I agree not? –  awllower May 27 '12 at 7:37

About the only proof of this result I have ever seen is the one using unique factorization in the quadratic domain $Z[\sqrt{-2}]$. Using infinite descent it is possible to determine all rational points on the elliptic curve, and showing that $(3,5)$ is the only integral point seems to require stuff like Baker's theorem. I have been looking for a proof that Fermat could have understood for years, and would be grateful if anyone could come up with one.

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