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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 :)2012-05-24
  • 7
    My proof won't fit in the margin.2012-05-24
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    I was able to check that $(3,5)$ is indeed a solution.2012-05-24
  • 3
    You should be careful to mention you want integer solutions.2012-05-24
  • 0
    Per chance the unique factorization in the extension $Q((-2)^{1/2})$ is acceptable?2012-05-24
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    $(3,-5)$ is also a solution2012-05-24
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    A solution using the fact that $\mathbb Z[\sqrt{-2}]$ is an [UFD](http://en.wikipedia.org/wiki/Unique_factorization_domain) is given in the book Titu Andreescu,Dorin Andrica,Ion Cucurezeanu: An Introduction To Diophantine Equations, [p.169](http://books.google.com/books?id=D_XmfolL-IUC&pg=PA169). However, this is not accessible to high school students.2012-05-24
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    Ah! Infinite descent is well-known already to high school students, right? If not, then why not explain to them? If so, then an approach might be acceptable, as shown in the book **Number Theory: An approach through history, from Hammurapi to Legendre** by ***A.Weil***. If I recall correctly.2012-05-24
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    @Martin Sleziak, very nice reference!2012-05-24
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    not only 5,but as well -5 is also solution2012-05-24
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    @Martin: the use of ${\mathbf Z}[\sqrt{-2}]$ is not acceptable to high school students? Gosh, I learned that proof when I was in high school... :)2012-05-24
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    @awllower I found solution of $y^2=x^3-2x$ using infinite descent in Weil's book, [p.150](http://books.google.com/books?id=d32SGbHnMKcC&pg=PA150). Is the solution of $y^2=x^3-2$ given there too?2012-05-24
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    I don't have the book handy, but I think this equation is solved in Uspensky & Heaslett, which is an introductory text in Number Theory. It is, by the way, an example of a "Mordell equation", and a websearch on that term will give you some idea of the difficulties involved. (Aside to @KCd, bet you learned it same place I did, in Columbus.)2012-05-24
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    This question also deals with the same equation: [How to find all rational points on the elliptic curves like $y^2=x^3-2$](http://math.stackexchange.com/questions/91437/how-to-find-all-rational-points-on-the-elliptic-curves-like-y2-x3-2).2012-05-24
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    I checked; U & H, Elementary Number Theory, pages 398 to 399. The proof is indeed via ${\bf Z}[\sqrt{-2}]$, but oddly never explicitly mentions use of unique factorization.2012-05-25
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    @MartinSleziak: Allow me, please, to put my hand on the book 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.

  • 8
    We're waiting...2012-05-24
  • 0
    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 http://www.eulersociety.org/ There are other master site?2012-05-25
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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