# JacobiSymbol

JacobiSymbol[n,m]

gives the Jacobi symbol .

# Details

• Integer mathematical function, suitable for both symbolic and numerical manipulation.
• For prime m, the Jacobi symbol reduces to the Legendre symbol. The Legendre symbol is equal to depending on whether n is a quadratic residue modulo m.
• JacobiSymbol automatically threads over lists.

# Examples

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## Basic Examples(2)

Compute Jacobi symbols:

Plot the sequence wrt the last argument:

## Scope(3)

Evaluate for large arguments:

Second argument may be even:

## Generalizations & Extensions(2)

JacobiSymbol threads element-wise over lists and arrays:

JacobiSymbol works for negative first arguments:

## Applications(5)

Find EulerJacobi pseudoprimes:

The Gauss reciprocity law:

Construct eigenvectors of the discrete Fourier transform:

Evaluate Gauss sums in closed form:

Plot the nontrivial values of the Jacobi symbol:

## Properties & Relations(1)

Reduce equations containing JacobiSymbol:

## Neat Examples(1)

Successive differences of JacobiSymbol modulo 2:

Wolfram Research (1988), JacobiSymbol, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiSymbol.html.

#### Text

Wolfram Research (1988), JacobiSymbol, Wolfram Language function, https://reference.wolfram.com/language/ref/JacobiSymbol.html.

#### CMS

Wolfram Language. 1988. "JacobiSymbol." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/JacobiSymbol.html.

#### APA

Wolfram Language. (1988). JacobiSymbol. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/JacobiSymbol.html

#### BibTeX

@misc{reference.wolfram_2023_jacobisymbol, author="Wolfram Research", title="{JacobiSymbol}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiSymbol.html}", note=[Accessed: 20-February-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2023_jacobisymbol, organization={Wolfram Research}, title={JacobiSymbol}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiSymbol.html}, note=[Accessed: 20-February-2024 ]}