Timeline for integer solutions to quadratic forms
Current License: CC BY-SA 2.5
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 30, 2010 at 3:01 | vote | accept | john mangual | ||
| Jul 29, 2010 at 22:35 | comment | added | Pietro Majer | Also (especially for rational solutions): make the substitutions w:=x+h, and the equation becomes linear in x. Use y,z,h as parameters and solve in x. Normalizing gives a parametrization for integers solution also (with some care about parity). | |
| Jul 29, 2010 at 22:33 | answer | added | Tony Scholl | timeline score: 15 | |
| Jul 29, 2010 at 22:14 | comment | added | Qiaochu Yuan | Matt's approach is the way I would go. It's worth noting that a similar approach works for Pythagorean triples and, with a little effort, for Pythagorean quadruples x^2 + y^2 + z^2 = w^2 as well. | |
| Jul 29, 2010 at 21:22 | comment | added | Daniel Litt | +1 Eric Tressler. Expanding on is comment, the relevant equations on the mathworld page are labeled 16-18. Basically, your question boils down to finding the splittings of the prime factors of $n=x^2+y^2$ that are equal to 1 mod 4. | |
| Jul 29, 2010 at 21:18 | comment | added | Matt Young | By rearranging your formula, you get to the equation $(x-z)(x+z)=(w-y)(w+y)$. Basically $x-z$ and $x+z$ can be viewed as independent variables (the slight catch is that they have the same parity) and similarly for the right hand side. So in effect you want to solve the equation $ab = cd$ where $a = b \pmod{2}$ and $c = d \pmod{2}$. By letting $a$ and $b$ vary you then want to find all the different factorizations of $ab$ with a constraint modulo $2$. | |
| Jul 29, 2010 at 21:08 | comment | added | Eric Tressler | A good starting point is research.att.com/~njas/sequences/A004018, which will tell you the number of ways to write $n$ as the sum of two squares, and give you some references. See also mathworld.wolfram.com/SumofSquaresFunction.html | |
| Jul 29, 2010 at 20:59 | history | asked | john mangual | CC BY-SA 2.5 |