Timeline for seeking an integer parameterization for A^2+B^2=C^2+D^2+1
Current License: CC BY-SA 3.0
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Aug 13, 2012 at 17:19 | comment | added | Igor Rivin | @Noam: I am fully aware of what you say, but notice that transforming this (essentially as you did, and as Matt Young did in the referenced question) into $(D+B)(D-B) = (A+C)(A-C) - 1$ allows one to generate all the solutions by letting $u=A+C, v = A-C$ and then $(D+B), (D-B)$ factors of $uv - 1$ (modulo some parity considerations) gives a quick way to generate solutions (but does not quite answer the question, I admit). | |
| Aug 13, 2012 at 16:58 | comment | added | Noam D. Elkies | I'm afraid it only looks similar: $A^2+B^2=C^2+D^2$ is a homogeneous equation, so solutions amount to rational points on a surface (up to scaling), which are easily parametrized in this case (the surface is birational to ${\bf P}^1 \times {\bf P}^1$; whereas $A^2+B^2=C^2+D^2+1$ is integral points on a smooth threefold, so the underlying geometry is quite different. | |
| Aug 13, 2012 at 16:06 | history | answered | Igor Rivin | CC BY-SA 3.0 |