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:

TraditionalForm formatting:

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:

Introduced in 1988
 (1.0)