Timeline for Integer solutions of $x^2=4+8y^2+13z^2$
Current License: CC BY-SA 3.0
9 events
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| Apr 21, 2015 at 21:51 | comment | added | GH from MO | Your problem is a special case of Theorem 6.2 in Chapter 13 of Cassels: Rational quadratic forms. This theorem says that there is an algorithm to find the reduced representations (whose number is finite), and every other representation is an image of a reduced one by an automorph of $-x^2+8y^2+13z^2$. So it remains to find the group of automorphs, for which Sections 4,5,7,8 of Chapter 13 in the book might be helpful. | |
| Apr 21, 2015 at 21:16 | comment | added | Mendes | Dear @GH, I have looked through your answer to mathoverflow.net/questions/114707/. However, this special case is not quite helpful for me! Could you please kindly post an answer describing an effective algorithm of determination of the orbits of the integral automorphism group of a quadratic form in the set of integer solutions in the general case, say on the example of the equation $x^2-8y^2-13z^2=4$? If necessary, I can learn $p$-adic numbers and adeles in order to understand your answer! | |
| Apr 21, 2015 at 7:18 | comment | added | Mendes | @GH, Thank you for your answer. Its good to know that it can be solved. I went also further considering the case (not important for my purposes) where the coefficients are $+1$ and $-1$, but not as further as you did. However, this case with coefficients $4, 8, 13$ appears to be more delicate. | |
| Apr 21, 2015 at 4:48 | comment | added | individ | In General, the solution of this equation is determined by the solutions of the Pell equations. Approximately in this form. math.stackexchange.com/questions/74931/… | |
| Apr 20, 2015 at 21:14 | history | edited | GH from MO |
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| Apr 20, 2015 at 20:59 | comment | added | GH from MO | One can parametrize all solutions. I recommend the first and last paragraph of my response to a similar question: mathoverflow.net/questions/114707/… | |
| Apr 20, 2015 at 20:49 | history | edited | Stanley Yao Xiao | CC BY-SA 3.0 |
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| Apr 20, 2015 at 20:17 | review | First posts | |||
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| Apr 20, 2015 at 20:16 | history | asked | Mendes | CC BY-SA 3.0 |