Timeline for Which Diophantine equations can be solved using continued fractions?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Oct 16, 2011 at 4:32 | comment | added | Samuel Hambleton | Thank you both very much, and sorry about my identity crisis. I initially wanted to vote the answer to Question 70913 to a non-negative number. I don't want to be deceitful. The incomplete "parametrization" I found for $x^2 - D y^2 = 4 z^3$ is $((t^3+3 D t u^2)/4, (3 t^2 u + D u^3)/4 , (t^2 - D u^2)/4)$ but I can't seem to get elements of to narrow class group of exact order $3$ with this "parametrization". I am also interested in solving $18 x y + x^2 y^2 - 4 x^3 - 4 y^3 - 27 = D z^2$. | |
| Oct 14, 2011 at 23:37 | comment | added | Kevin O'Bryant | With a change of variables, one can handle all of the equations $A x^2 + B x y + C y^2 + D x + E y + F =0$ in much the same way as Pell's Equation. | |
| Oct 14, 2011 at 20:02 | comment | added | Noam D. Elkies | @R.Thornburn: 1a) You're welcome! 1b) Perhaps Artin was thinking about point-counting on elliptic curves modulo a prime; see the paragraph I inserted. 2) Looks like this equation $x^2-Dy^2=4z^3$ will involve the $3$-torsion in the class group of ${\bf Z}[(D+\sqrt{D})/2]$, which may be accessible via the continued fraction for $(D+\sqrt{D})/2$ but I suspect that this is not the most efficient method for large $D$. | |
| Oct 14, 2011 at 19:59 | history | edited | Noam D. Elkies | CC BY-SA 3.0 |
Insert paragraph on Cornacchia and Schoof
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| Oct 13, 2011 at 3:40 | comment | added | Samuel Hambleton | This is a slightly different topic now but I've seen a cool paper of Professor Elkies' : "Pythagorean triples and Hilbert's Theorem 90", which I tried to apply to $x^2 - D y^2 = 4 z^3$. It partially worked but I couldn't seem to get the points I was interested in, and so I was wondering about continued fractions. | |
| Oct 13, 2011 at 2:51 | comment | added | Samuel Hambleton | Awesome. Thank you Professor Elkies. That should get me started. I can't remember who told me about continued fractions with respect to elliptic curves but I thought they mentioned Artin. I'm not sure. | |
| Oct 13, 2011 at 2:48 | vote | accept | Samuel Hambleton | ||
| Oct 13, 2011 at 2:42 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |