Fermat's proof of FLT(4) is an example of infinite descent as is Euler's (or whoever you attribute it to's) proof of FLT(3). There are similar proofs to Fermat's for Diophantine equations like $x^4 + y^4 = 2z^2$.

I have unsuccessfully tried to view these proofs in terms of group homomorphisms on conics and elliptic curves but it is not at all clear whether this is possible.

Can we reinterpret these infinite descent proofs geometrically, in terms of curves?

  • $\begingroup$ In the case of FLT(4), I was thinking of considering the conic $X^2 + Y^2 - 1$ and there is a subset of square points on it, "descent" would prove that the only square points are the trivial solutions. $\endgroup$
    – Quanta
    May 14 '11 at 1:26
  • $\begingroup$ Franz Lemmermeyer's nice articles Higher Descent on Pell Conics I, II, III may be of interest. $\endgroup$ Jun 13 '17 at 20:38

(That picture in your avatar is Weil, right? You should start by reading Weil's Number Theory: an approach through history).

FLT(3) is the assertion that the curve $x^3+y^3=1$ has three rational points (including the point at infinity). The standard process of putting an elliptic curve in Weierstrass form shows that this curve is $y^2=x^3-432$ if I remember correctly. Now use descent on this elliptic curve (maybe an isogeny of degree 3) to show that the Mordell-Weil group is finite of order three (see, e.g. Silverman for the general theory). This may be Euler's proof, maybe it's discussed in Weil's book.

FLT(4) is weaker than the statement that the equation $x^4+y^4=z^2$ has only the obvious solutions. Again, this becomes the problem of finding the rational points on $y^2=x^4+1$ which is again an elliptic curve and Fermat's proof is a 2-descent showing that the Mordell-Weil group is finite of order four. I'm pretty sure this is in Weil's book.

I have no idea what conics have to do with any of this.

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    $\begingroup$ I think the photo is of Schrödinger. $\endgroup$
    – S. Carnahan
    May 13 '11 at 22:46
  • $\begingroup$ According to Elkies math.harvard.edu/~elkies/nature.html#27 , Euler's proof for FLT(3) indeed amounts to a $3$-descent between elliptic curves of conductor 27. $\endgroup$ May 13 '11 at 23:10
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    $\begingroup$ Scott seems to be right about the photo. Incidentally, google pictures seems to be confused between Andre Weil, Andrew Weil (the alternative doctor) and Andrew Wiles. $\endgroup$ May 13 '11 at 23:12
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    $\begingroup$ (For FLT(4) see math.harvard.edu/~elkies/nature.html#32 and for FLT(7) see math.harvard.edu/~elkies/nature.html#49 ) $\endgroup$ May 13 '11 at 23:14
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    $\begingroup$ Conics come in when you construct the descent map for elliptic curves of the type $y^2 = x^4 + ax^2 + b$: one approach is looking at a parametrization of the conic $y^2 = z^2 + az + b$. Most elementary proofs of FLT4, for example, use the parametrization of the unit circle (Pythagorean triples). $\endgroup$ May 14 '11 at 12:04

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