# Questions tagged [legendre-symbol]

For questions involving the Legendre symbol, $\genfrac{(}{)}{}{}{a}{p}$ for integer $a$ and prime $p$.

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### Legendre symbol, second supplementary law

$$\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}$$ how did they get the exponent. May be from Gauss lemma, but how. Suppose we have a = 2 and p = 11. Then n = 3 (6,8,10), but not $$15 = (11^2-1)/8$$ n ...
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### If $p=a^2+4b^2$ for some $a,b \in \mathbb{Z}$, then $a$ is quadratic residu modulo $p$?

If $p=a^2+4b^2$ for some $a,b \in \mathbb{Z}$ and $p$ prime, then $a$ is quadratic residu modulo $p$? Approach: I thought it was true. (I could't find a counterexample). So I tried to prove it. I ...
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### Does the Legendre Symbol/quadratic reciprocity generalize to higher degrees?

The Legendre symbol is a tool for measuring whether or not $$x^2 \equiv a \text{ } (p)$$ has a solution in $\mathbb{F}_p$ for some fixed integer $a$. Does the Legendre symbol generalize to higher ...
125 views

### Connection between sgn character and the Legendre symbol

Today, while I was lecturing on the Legendre symbol, I realized that the phenomenon: "the product of two non-squares is a square" isn't so foreign. For example, for $\mathbb R^\times$, the squares are ...
377 views

### Prove that $-3$ is a quadratic residue mod an odd prime $>3$ if and only if $p$ is of the form of $6n+1$

How would I prove that $-3$ is a quadratic residue mod an odd prime larger than $3$ if and only if $p$ is of the form of $6n+1$? The last thing we covered in class last night was Euler criterion ...
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### Proof involving Legendre Symbol: $\left(\frac{3}{p}\right) = 1$ iff $p \equiv \pm 1 \pmod{12}$

I’m having a really difficult time with the following proof involving the Legendre symbol: Show that $\left(\dfrac{3}{p}\right) = 1$ iff $p \equiv \pm 1 \pmod{12}$ The normal tricks don’t seem to ...
### A problem with the Legendre/Jacobi symbols: $\sum_{n=1}^{p}\left(\frac{an+b}{p}\right)=0$ [duplicate]
This problem is taken from Niven's textbook, 3.6.16. Prove that if $(a,p)=1$ and $p$ is an odd prime, then $\sum_{n=1}^{p}\left(\frac{an+b}{p}\right)=0$, where $\left(\frac{x}{y}\right)$ is the ...
### Proof that $5$ is a quadratic residue $(\mod p)$ with $p$ odd prime iif $p \equiv \pm 1 \mod 10$
Here I present the following proof in order to receive corrections or any kind of suggestion to improve my handling/knowledge of modular arithmetic: Prove that $5$ is a quadratic residue $(\mod p)$ ...