I tried other similar question's solution on here but it doesn't work with this problem. What theorem should I use?
Asked
Active
Viewed 58 times
0
-
1Welcome to MSE. Your question is phrased as an isolated problem, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be closed. To prevent that, please [edit] the question. This will help you recognize and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Dec 31 '22 at 10:14
-
Is $3$ a quadratic residue modulo $2003$? – Dietrich Burde Dec 31 '22 at 10:40
-
I don't agree that this is a duplicate of the post it was linked to, but do consider the posts here and here. Basically if a prime modulus (here it is $2003$) is of the form $4k+3$ then there is a recipe. Once you solve the problem for modulus $2003$, solve it for the modulus $2003^2$ by representing $x=x_0+2003n$ where $x_0$ is a solution for modulus $2003$. – Dec 31 '22 at 14:50
-
(The "lifting" process to get a solution $\pmod{2003^2}$ from a solution $\pmod{2003}$ is called "Hensel's lemma" and the linked question has a link to an article explaining it in full generality.) – Dec 31 '22 at 14:56
-
(The "recipe" for solving $x^2\equiv a\pmod{p}$ is, by the way just checking if $a$ is a quadratic residue $\pmod{p}$, and, if yes (i.e. if $a^\frac{p-1}{2}\equiv 1\pmod{p}$), and if $p=4k+3$, the solution "just" comes out in the wash as $x\equiv \pm a^\frac{p+1}{4}\pmod{p}$.) – Dec 31 '22 at 15:01