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Fermat proved the Diophantine equation $$(x^2)^2 + (y^2)^2 = z^2$$ has only solutions $(0,0,0)$, $(0,\pm 1,\pm 1)$ and $(\pm 1,0,\pm 1)$ using "infinite descent".

The conic $C : X^2 + Y^2 - 1$ has a group law and rational points on this conic, with square numerator are exactly the solutions of the Diophantine equation.

I keep reading about how these descent arguments are related to isogeny of curves and related things but I have not been able to write them that way. Can the descent argument be explain in terms of the group law of the conic?

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