# Questions tagged [legendre-symbol]

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

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### Prove that $\sum_{X=0}^{p-1} \left(\frac{X^{2}+A}{p}\right)=-1$

Let $\left(\frac{a}{p}\right)$ denote the Legendre symbol and $p \geq 3$ a prime number and $A$ an integer with $p \nmid A$. Prove that $$\sum_{X=0}^{p-1} \left(\dfrac{X^{2}+A}{p}\right)=-1 .$$ I ...
1answer
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### Is there a quick way of using Gauss's lemma for large $p$?

Gauss's lemma in number theory states that if we have a number $a$ which is coprime to a prime $p$ then if we consider the set $S=\{a,2a,3a,.....((p-1)/2)a\}$ then say that if the number of elements ...
1answer
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### How to determine $(\tfrac{-5}{p})$ $\mod 20$?

There was a question in a past exam paper that asked Find criteria $\mod(20)$ for determining the Legendre symbol $(\tfrac{-5}{p})$ where $p\geq 7$. I am very confused by what this means as I ...
1answer
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### Legendre symbol, what is it?

I am reading wiki article about Legendre symbol and I don't understand the power meaning. Can you please explain the next expression. $$\left(\frac ap\right)\equiv a^{\frac{p-1}{2}}\pmod p$$
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1answer
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### Find all primes $p$ : $x^2 \equiv 13 \pmod p$ has solutions

I'm analyzing this little problem: Find all primes $p$ : $x^2 \equiv 13 \pmod p$ has solutions Here my effort since now: If the congruence $x^2 \equiv 13 \pmod p$ has solutions, must be the ...
0answers
352 views

### Determine the Legendre symbol of $\left(\frac{14}{p}\right)$

I have been asked to determine the Legendre symbol $\left(\frac{14}{p}\right)$ for $p \geq 11$ and have made good progress, however, I am stuck at the very last hurdle. So far, I have found that \...