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NumberTheory

  

KroneckerSymbol

  

generalized Jacobi symbol

  

JacobiSymbol

  

generalized Legendre symbol

  

LegendreSymbol

  

quadratic residuosity

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

KroneckerSymbol(a, n)

JacobiSymbol(a, m)

LegendreSymbol(a, m)

Parameters

a

-

integer

n

-

integer

m

-

positive odd integer

Description

• 

The KroneckerSymbol(a, n) command computes the Kronecker symbol of a and n.

• 

The alternative calling sequences, JacobiSymbol(a, m) and LegendreSymbol(a, m), have return values equal to KroneckerSymbol(a, m), but m must be a positive odd integer.

• 

The Legendre symbol is typically defined only for second arguments that are prime, but due to primality checking being expensive, here LegendreSymbol is an alias of JacobiSymbol.

• 

If n is equal to ui=1kpidi where u is −1 or 1 and the pi are distinct primes, then the Kronecker symbol an is given by aui=1kapidi, where ak is the usual Legendre symbol, except for the following cases.

a0 is equal to 1 if a is equal to 1 or −1. Otherwise it is equal to 0.

a-1 is equal to −1 if a is less than 0. Otherwise it is equal to 1.

a1 is always equal to 1.

a2 is equal to 2a if a is odd. Otherwise it is equal to 0.

Examples

withNumberTheory:

22 is congruent to 0 modulo 11.

LegendreSymbol22,11

0

(1)

9 is a quadratic residue modulo 11.

LegendreSymbol9,11

1

(2)

10 is a quadratic non-residue modulo 11.

LegendreSymbol10,11

−1

(3)

11 is congruent to 2 modulo 3 and is a quadratic non-residue modulo 3.

KroneckerSymbol11,32

1

(4)

KroneckerSymbol9,211

1

(5)

Compatibility

• 

The NumberTheory[KroneckerSymbol], NumberTheory[JacobiSymbol] and NumberTheory[LegendreSymbol] commands were introduced in Maple 2016.

• 

For more information on Maple 2016 changes, see Updates in Maple 2016.

See Also

NumberTheory