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NumberTheory

  

PseudoPrimitiveRoot

  

pseudo primitive root modulo n

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

PseudoPrimitiveRoot(n, options)

Parameters

n

-

positive integer

options

-

(optional) at most one of greaterthan = m or ith = i, where m is a non-negative integer and i is a positive integer

Description

• 

The PseudoPrimitiveRoot(n) command returns the smallest pseudo primitive root modulo n, if it exists.

• 

The PseudoPrimitiveRoot(n, greaterthan = m) command returns the smallest pseudo primitive root modulo n greater than m.

• 

The PseudoPrimitiveRoot(n, ith = i) command returns the ith smallest pseudo primitive root modulo n.

• 

If the required pseudo primitive root does not exist, then an error message is displayed.

• 

A pseudo primitive root is an integer y such that y and n are coprime, and there does not exist an integer x such that xr=ymodn where r is a divisor of n not equal to 1.

• 

If a primitive root modulo n exists, then the pseudo primitive roots are exactly the primitive roots.

Examples

withNumberTheory:

PseudoPrimitiveRoot7

3

(1)

PseudoPrimitiveRoot2662,greaterthan=2342

2345

(2)

There does not exist a primitive root modulo 8 but there are three pseudo primitive roots modulo 8.

PrimitiveRoot8

Error, (in NumberTheory:-PrimitiveRoot) there does not exist a primitive root modulo 8

seqPseudoPrimitiveRoot8,ith=i,i=1..3

3,5,7

(3)

An error message is displayed when the desired pseudo primitive root does not exist.

PseudoPrimitiveRoot8,ith=4

Error, (in NumberTheory:-PseudoPrimitiveRoot) there exist only 3 pseudo primitive roots modulo 8

PseudoPrimitiveRoot8,greaterthan=7

Error, (in NumberTheory:-PseudoPrimitiveRoot) there does not exist a pseudo primitive root modulo 8 greater than 7

Compatibility

• 

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

• 

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

See Also

NumberTheory

NumberTheory[MultiplicativeOrder]

NumberTheory[PrimitiveRoot]