# 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: Showing that $\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0$

I have a question about Legendre symbol. Let $a$, $b$ be integers. Let $p$ be a prime not dividing $a$. Show that the Legendre symbol verifies: $$\sum_{m=0}^{p-1} \left(\frac{am+b}{p}\right)=0.$$ I ...
2answers
3k views

### 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 ...
1answer
2k views

1answer
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### If $n\in\Bbb Z^+$ is not a square, prove exist infinitely many primes $p$ such that $\left(\frac{n}{p}\right)=-1$.

If $n\in\Bbb Z^+$ is not a square, prove exist infinitely many primes $p$ such that $\left(\frac{n}{p}\right)=-1$. Note that if $p\nmid n$ and $n=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_k^{\alpha_k}$, ...
2answers
144 views

### Find the primes $p$ such that the equation: $x^{2} + 6x + 15 = 0$ has a solution modulo $p$

I need to solve this question: Find the primes $p$ such that the equation: $x^{2} + 6x + 15 = 0$ has a solution modulo $p$. My approach was: I checked for $p = 2$ and there is no solution. Now ...
2answers
382 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 ...
1answer
671 views

2answers
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### Fast legendre symbol calculation

Let's say that I would like to calculate all legendre symbols from $1$ to $p-1$ $\pmod{p}$, is there a way to calculate them in an incremental way?. For example, an incremental table of legendre ...
1answer
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