Hi,

I am not too good at this stuff, but getting better haha..

Anyway, find an x which satisfies the statement 5003 divides x^2 + 10, or else prove that no such x exists.

I started by switching it around to say x^2 = -10 mod 5003.
I kind of suspect there is no solution, and there is a theorem (euler's criterion) which states that -10 is a quadratic residue (i.e. there is a solution) if and only if (-10)^(5002/2) = (-10)^2501 = 1 mod 5003.

(-10)^2501 is a huge number tho, and i'm not sure how to deal with it. Any hints would be great.

2. You need to use the Euler-Gauss quadratic reciprocity law, along with the fact that $\displaystyle \left(\frac{-10}{5003}\right)=\left(\frac{-1}{5003}\right)\left(\frac{2}{5003}\right)\left(\f rac{5}{5003}\right)$. (Assuming you are familiar with this notation for the Legendre symbol).
The text passage contains a mathematical question and a request for hints or further explanation, but no direct answer to the question is provided.
single_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Hi,

I am not too good at this stuff, but getting better haha..

Anyway, find an x which satisfies the statement 5003 divides x^2 + 10, or else prove that no such x exists.

I started by switching it around to say x^2 = -10 mod 5003.
I kind of suspect there is no solution, and there is a theorem (euler's criterion) which states that -10 is a quadratic residue (i.e. there is a solution) if and only if (-10)^(5002/2) = (-10)^2501 = 1 mod 5003.

(-10)^2501 is a huge number tho, and i'm not sure how to deal with it. Any hints would be great.

2. You need to use the Euler-Gauss quadratic reciprocity law, along with the fact that $\displaystyle \left(\frac{-10}{5003}\right)=\left(\frac{-1}{5003}\right)\left(\frac{2}{5003}\right)\left(\f rac{5}{5003}\right)$. (Assuming you are familiar with this notation for the Legendre symbol).
user
The text passage contains a mathematical question and a request for hints or further explanation, but no direct answer to the question is provided.

TYPE: SINGLE_QUESTION