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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

  • 8
    Well, it depends on the high school student :)2012-05-24
  • 7
    My proof won't fit in the margin.2012-05-24
  • 2
    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
  • 1
    $(3,-5)$ is also a solution2012-05-24
  • 2
    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
  • 0
    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
  • 0
    @Martin Sleziak, very nice reference!2012-05-24
  • 0
    not only 5,but as well -5 is also solution2012-05-24
  • 1
    @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
  • 0
    @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
  • 1
    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
  • 1
    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
  • 1
    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
  • 0
    @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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