KroneckerSymbol

The Kronecker symbol (a/n): the Jacobi symbol extended to every integer n.

✓ checked against that system · ! checked, with disagreements (hover a chip)FungrimKroneckerSymbol↗Wolfram LanguageKroneckerSymbol✓↗

KroneckerSymbol(a, n)the Kronecker symbol $\left(\frac{a}{n}\right)$.

SageMathkronecker(a, n)SymPykronecker_symbol(a, n)Oscarkronecker_symbol(ZZ(a), ZZ(n))Juliakronecker_symbol(ZZ(a), ZZ(n))

Typed signatures

Type	Package	Applies
(integer, integer) -> integer	modular	always

Domain: Number theory

Details

• Agrees with JacobiSymbol wherever $n$ is odd and positive, and with LegendreSymbol where $n$ is also prime — this is the head above both.
• $\left(\frac{a}{0}\right)=1$ if $a=\pm1$, else 0; $\left(\frac{a}{-1}\right)=1$ if $a\ge0$, $-1$ if $a<0$.
• $\left(\frac{a}{2}\right)=0$ for even $a$; $1$ if $a\equiv\pm1\pmod 8$; $-1$ if $a\equiv\pm3\pmod 8$ — this is the piece Jacobi's odd-$n$ restriction leaves out.
• Completely multiplicative in $n$: $\left(\frac{a}{n_1n_2}\right)=\left(\frac{a}{n_1}\right)\left(\frac{a}{n_2}\right)$.
• Bignum-safe: both arguments may exceed the double-precision range.

Plot

Examples

37 more kept as data, checked against the oracles

Basic #

live ↗

In$\mathrm{KroneckerSymbol}(17, 6)$Out$-1$

live ↗

In$\mathrm{KroneckerSymbol}(10, 13)$Out$1$

0 whenever |a| ≠ 1 and n = 0

live ↗

In$\mathrm{KroneckerSymbol}(5, 0)$Out$0$

(a/-1) is the sign of a — here a < 0

live ↗

In$\mathrm{KroneckerSymbol}(-3, -1)$Out$-1$

(a/2): odd a, and 3 mod 8 = 3, one of the two residues giving -1

live ↗

In$\mathrm{KroneckerSymbol}(3, 2)$Out$-1$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

live ↗

In$\mathrm{KroneckerSymbol}(\bigl\lbrack2, 3, 5, 7, 11\bigr\rbrack, 6)$Out$\bigl\lbrack0, 0, 1, 1, 1\bigr\rbrack$

Larger arguments: $10^{11}+1$ against the 2000th prime, 17389

live ↗

In$\mathrm{KroneckerSymbol}(10^{11}+1, \mathrm{NthPrime}(2\,000))$Out$-1$

Even $n$, where JacobiSymbol is undefined: $(7/2)(7/3)=1\cdot1$

live ↗

In$\mathrm{KroneckerSymbol}(7, 6)$Out$1$

A character periodic mod $7$ has a rational generating function with denominator of degree $6$

live ↗

In$\mathrm{GeneratingFunction}(\mathrm{KroneckerSymbol}(n, 7), n, x)$Out$\frac{x^5+2x^4+x^3+2x^2+x}{x^6+x^5+x^4+x^3+x^2+x+1}$

Applications #

The character mod 4: $(-4/p)=-1$ for primes $p\equiv3\pmod4$

live ↗

In$\mathrm{KroneckerSymbol}(-4, 7)$Out$-1$

Depends only on the residue mod $7$: $23 \equiv 2$

live ↗

In$\mathrm{KroneckerSymbol}(2, 7)=\mathrm{KroneckerSymbol}(23, 7)$Out$\mathrm{True}$

Threads over moduli; $-4$ is a square mod $1$ and $5$ but not mod $7$ or $11$

live ↗

In$\mathrm{KroneckerSymbol}(-4, \bigl\lbrack1, 5, 7, 11\bigr\rbrack)$Out$\bigl\lbrack1, 1, -1, -1\bigr\rbrack$

Properties #

Completely multiplicative in n

live ↗

In$\mathrm{KroneckerSymbol}(7, 3×5)=\mathrm{KroneckerSymbol}(7, 3)\mathrm{KroneckerSymbol}(7, 5)$Out$\mathrm{True}$

Zero whenever $\gcd(9,12) > 1$

live ↗

In$\mathrm{KroneckerSymbol}(9, 12)$Out$0$

Multiplicative in the top argument

live ↗

In$\mathrm{KroneckerSymbol}(12, 7)=\mathrm{KroneckerSymbol}(3, 7)\mathrm{KroneckerSymbol}(4, 7)$Out$\mathrm{True}$

Multiplicative in the bottom argument: $63 = 9\cdot7$

live ↗

In$\mathrm{KroneckerSymbol}(3, 63)=\mathrm{KroneckerSymbol}(3, 9)\mathrm{KroneckerSymbol}(3, 7)$Out$\mathrm{True}$

Possible issues #

$(a/0)=1$ for $a=\pm1$

live ↗

In$\mathrm{KroneckerSymbol}(-1, 0)$Out$1$

About

Examples	54
Checked against	Wolfram Language 54 agreeSageMath 53 agreeSymPy 53 agreeOscar 50 agreeJulia 52 agree
Crosswalk	FungrimWolfram LanguageSageMathSymPyOscarJulia

Implementation

mappedoscarpackages/oracle/src/mappings.ts

mappedwolframpackages/oracle/src/mappings.ts

mappedsympypackages/oracle/src/mappings.ts

kronecker_symbol is SymPy's extension of the Jacobi symbol to every integer n.

mappedsagepackages/oracle/src/mappings.ts

mappedjuliapackages/oracle/src/mappings.ts

See also: JacobiSymbol, LegendreSymbol