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NumberTheory

  

MultiplicativeOrder

  

order of a number under modular multiplication

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

MultiplicativeOrder(m, n)

Parameters

m

-

positive integer

n

-

positive integer greater than 1

Description

• 

The MultiplicativeOrder function computes the multiplicative order of m modulo n, which is defined as the least positive integer exponent i such that m^i is congruent to 1 modulo n.

• 

Alternatively, the multiplicative order can be defined as the order of the cyclic group generated by m under multiplication modulo n.

• 

The multiplicative order exists if and only if m and n are coprime. In the case that it does not exist, an error message is displayed.

Examples

withNumberTheory:

MultiplicativeOrder7,18

3

(1)

seq7imod18,i=1..3

7,13,1

(2)

If the multiplicative order of m is equal to the totient of n, then m is a primitive root modulo n.

Totient18

6

(3)

MultiplicativeOrder11,18

6

(4)

PrimitiveRoot18,greaterthan=10

11

(5)

Since 5 and 25 are not coprime, the multiplicative order of 5 modulo 25 is not defined.

MultiplicativeOrder5,25

Error, (in NumberTheory:-MultiplicativeOrder) the arguments 5 and 25 are not coprime

Compatibility

• 

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

• 

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

See Also

NumberTheory