Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [legendre-symbol]

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

In number theory, the Legendre symbol is a multiplicative function with values $1, −1, 0$ that is a quadratic character modulo a prime number $p$: its value on a (nonzero) quadratic residue mod p is 1 and on a non-quadratic residue (non-residue) is $−1$. Its value on zero is $0$.

The Legendre symbol was introduced by Adrien-Marie Legendre in 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet characters of higher order. The notational convenience of the Legendre symbol inspired introduction of several other "symbols" used in algebraic number theory, such as the Hilbert symbol and the Artin symbol.

Source: https://en.wikipedia.org/wiki/Legendre_symbol

299 questions
19
votes
3 answers

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 is a way to compute Legendre symbols from Gauss…
Yola
  • 1,763
11
votes
1 answer

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 degrees? For example, can I define a law $$ \left(…
54321user
  • 3,383
8
votes
0 answers

Application of Legendre, Jacobi and Kronecker Symbols

Legendre, Jacobi and Kronecker Symbols are powerful multiplicative functions in computational number theory. They are useful mathematical tools, essentially for primality testing and integer factorization; these in turn are important in…
Ali Abbasinasab
  • 1,783
7
votes
1 answer

Name of group of order $p-\bigl(\frac ap\bigr)$ constructed from field $\Bbb Z_p$?

Let $p$ be an odd prime, and $a$ be an element of field $\Bbb Z_p$. Define $l$ as the Legendre symbol $\displaystyle\biggl(\frac ap\biggr)$. When $l=+1$, define $b$ as a particular solution of $b^2=a$ in $\Bbb Z_p$, e.g. the odd one in range…
fgrieu
  • 1,828
7
votes
2 answers

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 where it has a quadratic residue if $a^{(p-1)/2}\equiv…
andemw01
  • 207
7
votes
2 answers

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 deduced that $a$ is a quadratic residu modulo $p$ iff…
bob
  • 1,276
7
votes
1 answer

How to prove this formula for the Legendre symbol for a finite field

Let $\mathbb{F}_q$ be a finite field with $q$ odd, let $x\in\mathbb{F}_q$ and define the Legendre symbol for $\mathbb{F}_q$ as \begin{equation} \left(\frac{x}{\mathbb{F}_q} \right) = \begin{cases} \phantom{-}1 & \text{if $t^2=x$ has a solution…
Kevin
  • 159
  • 5
6
votes
2 answers

How can I prove these summations for the legendre symbol

How can I prove for the Legendre symbol that: $$\sum_{a=1}^{p-1}{\left(\frac{a(a+1)}{p}\right)}= -1 = \sum_{b=1}^{p-1}{\left(\frac{(1+b)}{p}\right)}$$
kiki17
  • 61
  • 2
6
votes
1 answer

A prime number is not a quadratic residue modulo some prime without quadratic reciprocity

In Cox's book "Primes of form $x^2 + ny^2$", I stumbled upon a lemma $ \newcommand{\Z}{\mathbb{Z}} $ Lemma 1.14: If $D \equiv 0,1 \pmod{4}$ is a nonzero integer, then there is a unique homomorphism $\chi:(\Z/D\Z)^* \longrightarrow \{\pm 1\}$ such…
Kolja
  • 3,033
6
votes
2 answers

How to prove this sum related to Legendre symbol

I can see why it is true by writing out some examples, but I'm not sure how one could prove that, with $\left({\cdot\over p}\right)$ as the mod $p$ Legendre symbol $$\sum_{x=1}^{p-1} \left(\frac{x(x-1)}{p}\right)=-1$$ For example, for…
Quality
  • 5,807
  • 8
  • 37
  • 76
6
votes
1 answer

$\tau( n!(n+1) ) = 2\times\tau(n!)$. Then what is $n!\;\text{mod} \; (n+1)?$

If the number of factor of $(n+1)!$ is double than the number of factor of $n!$, then find the reminder if $n!$ is divided by $(n+1)$? I'm not sure if the question mean factor = divisor. However in both cases I cant find a way to start. Source:…
Rezwan Arefin
  • 3,216
6
votes
1 answer

How to solve $1+\frac12-\frac13+\frac14-\frac15-\frac16+\frac18+\ldots+\left(\frac n7\right)\frac1n+\ldots$?

$$1+\frac12-\frac13+\frac14-\frac15-\frac16+\frac18+\ldots+\left(\frac n7\right)\frac1n+\ldots$$ where $\left(\frac n7\right)$ is Legendre symbol. I think its about algebraic number theory, but I can't find relative problem on book. If the signs…
卓永鴻
  • 81
6
votes
2 answers

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 if $p \neq 2 $ so the equation has a solution…
CnR
  • 2,061
6
votes
0 answers

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 just $\mathbb R^{>0}$. Additionally, two…
Spooky
  • 751
6
votes
3 answers

When $p=3 \pmod 4$, show that $a^{(p+1)/4} \pmod p$ is a square root of $a$

Let $a$ and $p$ be integers such that $p$ is prime, and $a$ is a square modulo $p$. When $p\equiv3\pmod4$, show that $a^{(p+1)/4}\pmod p$ is a square root of $a$. Why does this technique not work when $p\equiv1\pmod4$? This is a question that…
dessskris
  • 335
1
2 3
…
19 20