$Quadratic residue$

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$Quadratic residue$

A quadratic residue (mod p) refers to an integer c such that there exists an integer x for which the congruence $$ x^2 \equiv c (mod p)$$ holds, right?
Rewriting:
$$x^2(mod p) \equiv c (mod p)$$
Which means that remainder of x^2 divided by p is the same as remainder of c divided by p.

Why c is called a residue? Residue of what?

Number Theory

Babaduras

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In modular arithmetic, when we say c is a quadratic residue (mod p), we mean that there exists an integer x such that when x^2 is divided by p, the remainder is c. Mathematically, this can be expressed as what you wrote.

In this context, c is referred to as a residue because it represents what remains after the division of x^2 by p. So, c is a residue of x^2 modulo p.

When we talk about quadratic residues (mod p), we're essentially exploring the residues that arise from squaring integers x and taking the remainders modulo p.

If c is between zero and p, then c is the residue. If c is greater than p, c (mod p) gives the residue.

Example:

$x^{2 }(mod 5) \equiv 9$

$9(mod 5) \equiv 4$

Here, 4 is the residue.

Kav10

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Babaduras

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But in the case when c > p, it can’t be called residue since it is not, it will leave some residue, but it itself is not, right?
- Kav10
  
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  That is correct. It will create the residue if you do one more mod as shown in the example.

$Quadratic residue$

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