Timeline for Solutions to nonhomogeneous quadratic equation mod $N$
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Sep 26, 2020 at 9:58 | comment | added | Maciej Ulas | @Gautam My parametrization is invertible over $\mathbb{Q}$ but not necessarily modulo $N$. Indeed, the inverse is given by $u=y/(x-1), v=z/(x-1)$ (here $z=(x^2+y^2-x)/N$) and can be computed modulo $N$ provided that $x-1$ is coprime to $N$. | |
| S Sep 25, 2020 at 1:03 | history | suggested | Gautam | CC BY-SA 4.0 |
Corrected some typos and simplified some sentences.
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| Sep 24, 2020 at 22:36 | comment | added | Gautam | Thanks, this is now clear. Does this capture all solutions? Intuitively, the family of solutions should depend on only one parameter, since there are two unknowns ($x$ and $y$) and one constraint. This leads me to believe that your proposal indeed captures all possible solutions, but I'm not sure how to prove it. | |
| Sep 24, 2020 at 22:31 | review | Suggested edits | |||
| S Sep 25, 2020 at 1:03 | |||||
| Sep 24, 2020 at 8:45 | comment | added | Maciej Ulas | @Gautam Of course you are right. I was to quick. I edited the answer and believe that everything is clear now. | |
| Sep 24, 2020 at 8:44 | history | edited | Maciej Ulas | CC BY-SA 4.0 |
added 78 characters in body
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| Sep 24, 2020 at 6:36 | comment | added | Gautam | I don't quite follow your answer. I verified that your choice $(x, y)$ is indeed a solution to the equation for $N = 1234567$. Can you explain what you mean by "using the trivial solution $x = 1, y = 0, z = 0$? How did you use this solution? Also, if we take $y = ux$, $z = vx$, we obtain the equation $x^2 + u^2x^2 - x - Nvx = 0$, whose solution is $x = \frac{vN + 1}{u^2 + 1}$, which is not the solution you described... I think your answer might have some small typos. I'm really interested to hear what you figured out, though! | |
| Sep 23, 2020 at 5:29 | history | edited | Maciej Ulas | CC BY-SA 4.0 |
added 2 characters in body
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| Sep 23, 2020 at 5:20 | history | answered | Maciej Ulas | CC BY-SA 4.0 |