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JacobiSymbol[n,m]

gives the Jacobi symbol .

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Generalizations & Extensions  
Applications  
Properties & Relations  
Neat Examples  
See Also
Tech Notes
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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:

Wolfram Language code: JacobiSymbol[10, 5]
Wolfram Language code: Table[JacobiSymbol[n, m], {n, 0, 10}, {m, 1, n, 2}]

Plot the sequence wrt the last argument:

Wolfram Language code: DiscretePlot[JacobiSymbol[100, n], {n, 50}]

Scope  (3)

Evaluate for large arguments:

Wolfram Language code: JacobiSymbol[10 ^ 10 + 1, Prime[1000]]

Second argument may be even:

Wolfram Language code: JacobiSymbol[7, 6]

TraditionalForm formatting:

Wolfram Language code: JacobiSymbol[n, k]//TraditionalForm

Generalizations & Extensions  (2)

JacobiSymbol threads element-wise over lists and arrays:

Wolfram Language code: JacobiSymbol[{2, 3, 5, 7, 11}, 3]

JacobiSymbol works for negative first arguments:

Wolfram Language code: JacobiSymbol[-3, {1, 3, 5, 7}]

Applications  (5)

Find Euler–Jacobi pseudoprimes:

Wolfram Language code: Select[Range[10000], GCD[#, 2] === 1 && Not[PrimeQ[#]] && JacobiSymbol[2, #] == Mod[2 ^ ((# - 1) / 2), #]&]

The Gauss reciprocity law:

Wolfram Language code: JacobiSymbol[p, q]JacobiSymbol[q, p] == (-1) ^ ((p - 1) / 2 (q - 1) / 2)
Wolfram Language code: Table[If[GCD[p, q] == 1, %, True], {p, 3, 10, 2}, {q, 3, 10, 2}] //Flatten//Union

Construct eigenvectors of the discrete Fourier transform:

Wolfram Language code: With[{p = 13}, Table[JacobiSymbol[k, p], {k, 0, p - 1}]]//N
Wolfram Language code: Fourier[%]//Chop

Evaluate Gauss sums in closed form:

Wolfram Language code: With[{h = 4, p = 11}, {Underoverscript[∑, k = 1, p]If[GCD[k, p] == 1, E^(2 π I h k^2/p), 0], JacobiSymbol[h, p] I^((p - 1/2))^2 Sqrt[p] - 1}]//FullSimplify

Plot the nontrivial values of the Jacobi symbol:

Wolfram Language code: ArrayPlot[Table[JacobiSymbol[n, m], {n, 0, 40}, {m, 1, 2n, 2}], ColorRules -> {1 -> Red, -1 -> Blue}]

Properties & Relations  (1)

Reduce equations containing JacobiSymbol:

Wolfram Language code: Reduce[JacobiSymbol[p, 3] == 2JacobiSymbol[p + 1, 3] - 1 && -5 < p < 5, p, Integers]

Neat Examples  (1)

Successive differences of JacobiSymbol modulo 2:

Wolfram Language code: ArrayPlot[Mod[NestList[Differences, JacobiSymbol[Range[100], 77], 100], 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_2026_jacobisymbol, author="Wolfram Research", title="{JacobiSymbol}", year="1988", howpublished="\url{https://reference.wolfram.com/language/ref/JacobiSymbol.html}", note=[Accessed: 12-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_jacobisymbol, organization={Wolfram Research}, title={JacobiSymbol}, year={1988}, url={https://reference.wolfram.com/language/ref/JacobiSymbol.html}, note=[Accessed: 12-June-2026]}