# Number theory - quadratic residues

• yeland404
In summary, a quadratic residue in number theory is an integer that is congruent to a perfect square modulo a given modulus. They are significant in providing information about the structure and properties of numbers, and have applications in cryptography, coding theory, and other fields of mathematics. To calculate quadratic residues, one can use mathematical functions like the Legendre or Jacobi symbols, or use theorems like quadratic reciprocity. It is possible for a number to be a quadratic residue for some moduli but not others, depending on the properties of the modulus. There are also several open problems related to quadratic residues, such as the existence of a primitive root modulo a composite number and the generalized Riemann hypothesis. These problems remain unsolved in various areas
yeland404

## Homework Statement

find all incongruent solutions of each quadratic congruence below.
X^2$\equiv$23 mod 77

## Homework Equations

X^2$\equiv$11 mod 39

## The Attempt at a Solution

it is suffices to X^2$\equiv$23 mod 7, andX^2$\equiv$23 mod 11,
then how to do next?

## 1. What is a quadratic residue in number theory?

A quadratic residue in number theory is an integer that is congruent to a perfect square modulo a given modulus. In other words, it is a number that, when squared, gives a remainder of a certain number when divided by the modulus.

## 2. What is the significance of quadratic residues?

Quadratic residues are important in number theory because they provide information about the structure and properties of numbers. They also have applications in cryptography, coding theory, and other areas of mathematics.

## 3. How do I calculate quadratic residues?

To calculate quadratic residues, you can use the Legendre symbol or the Jacobi symbol. These symbols are mathematical functions that determine whether a number is a quadratic residue or not. Alternatively, you can use quadratic reciprocity and other theorems to determine the quadratic residues of a given number.

## 4. Can a number be a quadratic residue for some moduli but not others?

Yes, a number can be a quadratic residue for some moduli but not others. This is because the quadratic residues depend on the properties of the modulus, such as its primality and congruence conditions. For example, a number may be a quadratic residue modulo a prime number but not a composite number.

## 5. Are there any open problems related to quadratic residues?

Yes, there are several open problems related to quadratic residues in number theory. For instance, the existence of a primitive root modulo a composite number is an open problem, and there is also the generalized Riemann hypothesis, which deals with the distribution of quadratic residues. Additionally, there are many unsolved problems related to quadratic residues in other areas of mathematics, such as cryptography and coding theory.

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