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March 9th, 2015, 09:51 AM   #1
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[SOLVED] Questions about QRs and Legendre symbol

Textbook: A Friendly Introduction to Number Theory 4th by Silverman

When $\displaystyle p$ is prime,
$\displaystyle a\overwithdelims () p$ = $\displaystyle \begin{cases} 1 & \text{if } a \text{ is a QR modulo p,} \\ -1 & \text{if } a \text{ is a NR modulo p} \end{cases}$

For me, QRs (Quadratic Residues) are numbers whose squares modulo p are perfect squares. For example,

0 -> 0
1 -> 1
2 -> 4
3 -> 4, since 9 modulo 5 is congruent to 4
4 -> 1, since 16 modulo 5 is congruent to 1

There's a theorem that states that if p is an odd prime, then there are exactly $\displaystyle (p-1)/2$ QRs modulo p and exactly $\displaystyle (p-1)/2$ NRs modulo p.

Doesn't the example above show 4 QRs?

EDIT:

My definition shows that I didn't understand quadratic residues at all. In the list I have, the QRs modulo 5 are 1 and 4, which makes 2 and 3 NRs.

Last edited by MadSoulz; March 9th, 2015 at 10:15 AM. Reason: Answered my own question
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March 9th, 2015, 10:53 AM   #2
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There are two kinds of squares mod a given odd prime: those which are coprime to the prime (this is 0) and the rest. There are (p-1)/2 nonresidues, (p-1)/2 coprime residues, and 0. The Legendre symbol tells you into which camp a given number falls.

Note that if you use the Jacobi symbol (n may be composite) the cases are less clear. There are noncoprime residues not equal to 0, and some nonresidues have Jacobi symbol 1. (But no residue has symbol -1.)
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March 10th, 2015, 07:09 AM   #3
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Quote:
Originally Posted by MadSoulz View Post
When $\displaystyle p$ is prime,
$\displaystyle a\overwithdelims () p$ = $\displaystyle \begin{cases} 1 & \text{if } a \text{ is a QR modulo p,} \\ -1 & \text{if } a \text{ is a NR modulo p} \end{cases}$
This assumes that $a$ is not divisible by $p$. Some authors add that $\displaystyle\left(\frac ap\right)=0$ iff $p\mid a$.
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