gives the Kronecker symbol .
- KroneckerSymbol is also known as the Jacobi symbol or Legendre symbol.
- Integer mathematical function, suitable for both symbolic and numerical manipulation.
- KroneckerSymbol[n,1] gives 1.
- KroneckerSymbol[n,-1] gives 1 whenever n is non-negative and otherwise.
- For a number with a unit and primes, returns .
Examplesopen allclose all
Basic Examples (2)
Plot the KroneckerSymbol sequence with respect to the second argument:
Numerical Manipulation (3)
Basic Applications (2)
Number Theory (9)
For congruent integers m and n modulo p, KroneckerSymbol[m,p]==KroneckerSymbol[n,p]:
Find Euler–Jacobi pseudoprimes to base : [more info]
The congruence equation has a solution if KroneckerSymbol[a,p] == 1:
A real primitive character χ modulo k can be written in terms of KroneckerSymbol[χ[-1]k,n]:
Nonprimitive real characters can be written in terms of KroneckerSymbol at integers coprime to k:
KroneckerSymbol is the generalization of the Jacobi symbol for all integers:
Properties & Relations (5)
KroneckerSymbol gives for non-coprime integers:
KroneckerSymbol is a completely multiplicative function for each argument: