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NumberTheory

  

RootsOfUnity

  

modular roots of unity

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

RootsOfUnity(k, n)

Parameters

k

-

prime number

n

-

positive integer

Description

• 

The RootsOfUnity(k, n) command computes all the kth roots of unity modulo n.

• 

An integer x is said to be a kth root of unity modulo n if xk=1modn.

Examples

with(NumberTheory):

unity := RootsOfUnity(5, 8965);

unity1,1631,2446,3261,6521

(1)

map(x -> x^5 mod 8965, unity);

1

(2)

Distribution of the second roots of unity. A point x,y on the plot denotes that y is a second root of unity modulo x.

plots:-pointplot(select(p -> p[2]^2 mod p[1] = 1, [seq(seq([i,j], j=0..(i-1)), i = 1..1000)]), labels=["Modulus", "Second roots of unity"], labeldirections=[horizontal, vertical]);

Compatibility

• 

The NumberTheory[RootsOfUnity] command was introduced in Maple 2016.

• 

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

See Also

NumberTheory