Jacobi Symbol - Maple Help
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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  where  is  or  and the  are distinct primes, then the Kronecker symbol  is given by , where  is the usual Legendre symbol, except for the following cases.

 is equal to  if  is equal to  or . Otherwise it is equal to .

 is equal to  if  is less than . Otherwise it is equal to .

 is always equal to .

 is equal to  if  is odd. Otherwise it is equal to .

Examples

 is congruent to  modulo .

(1)

 is a quadratic residue modulo .

(2)

 is a quadratic non-residue modulo .

(3)

 is congruent to  modulo  and is a quadratic non-residue modulo .

(4)

(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

 


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