Timeline for Intuition for the last step in Serre's proof of the three-squares theorem
Current License: CC BY-SA 2.5
4 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2023 at 14:59 | comment | added | KConrad | And the tangent line method would never work on a circle anyway (the context of the proof in Serre's book) since the tangent line to a point on a circle has no additional point of intersection with the circle. | |
| Oct 30, 2023 at 14:22 | comment | added | KConrad | It's not just with tangent lines: on $y^2 = x^3 + 17$, the line through $(2,5)$ and $(-2,3)$ meets the curve in the point $(1/4,33/8)$. As I had written in my answer, "intuition is one thing and checking the details is another: [...] the math has to work out to show the denominators really get smaller in the second solution you produce." Sometimes it works, sometimes not. I'm reminded of a proof that $\mathbf Z[i]$ is Euclidean where the method extends to $\mathbf Z[\sqrt{-2}]$, $\mathbf Z[\sqrt{2}]$. and $\mathbf Z[\sqrt{3}]$, but not $\mathbf Z[\sqrt{6}]$ despite being Euclidean. Oh well. | |
| Nov 2, 2009 at 12:18 | history | edited | Kevin Buzzard | CC BY-SA 2.5 |
improved clarity
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| Nov 2, 2009 at 11:42 | history | answered | Kevin Buzzard | CC BY-SA 2.5 |