Evaluate the Legendre symbol ## (999|823) ##

[Math100]

Homework Statement

Evaluate the Legendre symbol $(999|823)$.
(Note that $823$ is prime.)

Relevant Equations

Let $p$ be an odd prime. If $n\not\equiv 0\pmod {p}$, we define Legendre's symbol $(n|p)$ as follows:
$(n|p)=+1$ if $nRp$, and $(n|p)=-1$ if $n\overline{R}p$.
If $n\equiv 0\pmod {p}$, we define $(n|p)=0$.

Consider $(999|823)$.
Then $999\equiv 176\pmod {823}$.
This implies $(999|823)=(176|823)=(16|823)(11|823)=(4^{2}|823)(11|823)$.
Since $(a^{2}|p)=1$, it follows that $(4^{2}|823)=1$.
Thus $(999|823)=(11|823)$.
Applying the Quadratic reciprocity law, we have that
$(11|823)(823|11)=(-1)^{(11-1)(823-1)/4}=(-1)^{2055}=-1$.
Therefore, $(999|823)=-1$.

 

[fresh_42]

Homework Statement:: Evaluate the Legendre symbol $(999|823)$.
(Note that $823$ is prime.)
Relevant Equations:: Let $p$ be an odd prime. If $n\not\equiv 0\pmod {p}$, we define Legendre's symbol $(n|p)$ as follows:
$(n|p)=+1$ if $nRp$, and $(n|p)=-1$ if $n\overline{R}p$.
If $n\equiv 0\pmod {p}$, we define $(n|p)=0$.

Consider $(999|823)$.
Then $999\equiv 176\pmod {823}$.
This implies $(999|823)=(176|823)=(16|823)(11|823)=(4^{2}|823)(11|823)$.
Since $(a^{2}|p)=1$, it follows that $(4^{2}|823)=1$.
Thus $(999|823)=(11|823)$.
Applying the Quadratic reciprocity law, we have that
$(11|823)(823|11)=(-1)^{(11-1)(823-1)/4}=(-1)^{2055}=-1$.
Therefore, $(999|823)=-1$.

Looks good, although the notation is odd and I can only guess what $nRp$ means.

With Euler's criterion, we get
\begin{align*}
\left(\dfrac{999}{823}\right)&=\left(\dfrac{176}{823}\right)=176^{411}=\ldots = 822 = -1 \pmod{823}
\end{align*}

 

[Math100]

Looks good, although the notation is odd and I can only guess what $nRp$ means.

With Euler's criterion, we get
\begin{align*}
\left(\dfrac{999}{823}\right)&=\left(\dfrac{176}{823}\right)=176^{411}=\ldots = 822 = -1 \pmod{823}
\end{align*}

I think $nRp$ means that $n$ is a quadratic residue mod $p$. And $n\overline{R}p$ if $n$ is a quadratic nonresidue mod $p$. But how did you get $176^{411}=...=822$?

 

[fresh_42]

I think $nRp$ means that $n$ is a quadratic residue mod $p$. And $n\overline{R}p$ if $n$ is a quadratic nonresidue mod $p$.

Sure. I know. But the acronym is unusual.

But how did you get $176^{411}=...=822$?

I got it with WA but you have solved such equations before. $411=3\cdot 137$ should help a bit.