# Questions tagged [legendre-symbol]

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

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### Why does $a^n \mod p$ always result in a number with Legendre symbol as 1?

I noticed that the following expression $a^n\mod p$ where p is a prime and $n >=1$ and $n <= p$ always results in a number with Legendre Symbol (with p as the prime) as 1. I tested it out with a ...
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### Sum of Legendre Symbols for evey number less than $p$

If $p$ is an odd prime, show that $\sum_{j=1}^{p-1}({\frac{j}{p}})=0$ with $(\frac{j}{p})$ the Legendre Symbol. I'm not sure if it's enough to say that, because of the Euler's Criteria, there are ...
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### Squares in a finite field $\mathbb{F}_p$

I need to find all prime $p$ s.t $n+3$ is the inverse of $n-3$ in $\mathbb{F}_p$. So obviously this means $(n+3)(n-3)=1\mod p$, meaning $n^2=10\mod p$. So the question is - for which $p$ does the ...
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### Show for any odd prime $p\geq 5,$ $(-3/p)=1$ or $-1$ [duplicate]

Show for any odd prime $$p\geq 5,$$ $$\left ( \frac{-3}{p} \right ) =\begin{cases} 1 & \text{ if } p\equiv 1,-5\pmod{12} \\ -1& \text{ if } p\equiv -1,5\pmod{12} \end{cases}$$ So far I have ...
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### Investigate the size of $\sum_{x=0}^{p-1}e^{2\pi ix^2/p}$

I attempted to look at the size of this sum, and the hint was to use Legendre symbols to split this sum into 2 sums. But from what I have seen so far, all I manage to do is simplify the sum rather ...
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### Testing for primitive roots using quadratic non residue and Jacobi symbol

Is this always true for all cases?? $a$ is a primitive root $modulo$ $n$ $⇒$ $\left(\dfrac{a}{n}\right) = -1$ Is the converse also always true? $\left(\dfrac{a}{n}\right)$ $= -1$ $⇒$ $a$ is a ...
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### How to evaluate Legendre symbols more quickly

I'm trying to determine for which primes $p$ we have $\displaystyle\binom{6}{p}=1$, where $\displaystyle\binom{6}{p}$ is the the Legendre symbol. I know how to do this, but my question what is the ...
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### Proving Diophantine Equation has no solution using Legendre Symbol

Given that $\left(\frac{10}{23}\right)=-1$. How would I go about showing that $9x^2-46(y^3+3y+1)=10$ has no integer solutions? I believe it has something to do with Quadratic Reciprocity. For ...
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### Sum of Legendre symbols is 0?

I have a question regarding this sum: $$\sum_{k=1}^{p-1}k\left(\frac{k}{p}\right)$$ where $(k/p)$ is the Legendre symbol mod $p$, for $p>3$. I shall prove that \begin{...
### Cardinality of $S = \{ x \in \mathbb{Z}_p^* | \phi(1-x^2) = 1 \}$
For prime $p$, let, $S = \{ x \in \mathbb{Z}_p^* | \phi(1-x^2) = 1 \}$, where $p=4k+1$ and $\phi$ is Legendre symbol. I have to prove that $|S| = 2(k-1)$. I know that there are $(p-1)/2$ residues ...
### Does $x^4 \equiv -17 \pmod{83}$ has root or not?
I need to answer a question "Does $x^4 \equiv -17 \pmod{83}$ has root or not?" Here is my answer. We first prove that $X^2 \equiv -17 \pmod{83}$ has no root by using Legendre symbol. Indeed, \$\left( \...