Prove that $x ^ 3-y ^ 2 = 2$ only has one solution $(3,5)$

Question

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?

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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.

Answer 1

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.

Answer 2

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.

Answer 3

There is a completely elementary solution — accessible to any reasonably advanced high school math student — due essentially to Stan Dolan. See this answer to a duplicate question.