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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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    @Marti$n$Slezia$k$: Allow me, please, to put my hand on the boo$k$ again, so as to verify my previous assertion. Apology for any inconvenience.2012-05-27

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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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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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    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?2012-05-27