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NumberTheory

  

Totient

  

Euler's totient function

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

Totient(n)

 

phi(n)

φn

varphi(n)

ϕn

Parameters

n

-

positive integer

Description

• 

The Totient function computes Euler's totient function.

• 

Given a positive integer n, Totient(n) returns the number of positive integers coprime to n and not greater than n.

• 

phi and varphi are aliases of Totient.

• 

You can enter the commands phi and varphi using either the 1-D or 2-D calling sequence. For example, phi(8) is equivalent to φ8, and varphi(8) is equivalent to ϕ8.

Examples

withNumberTheory:

Totient1,φ2,ϕ3

1,1,2

(1)

If two integers n and m are coprime, then Totient(n)*Totient(m) = Totient(m*n). That is, the totient function is multiplicative.

Totient44Totient79,Totient4479

1560,1560

(2)

andmapi→igcdi1,i21xorφi1i2=φi1φi2,seqseqi,j,j=1..100,i=1..100

true

(3)

The totient of any prime p is equal to p-1.

Totient59,Totient101

58,100

(4)

andmapi→ithprimei1=ϕithprimei,seq1..100

true

(5)

The following command plots the values of Totient(n) for n from 2 to 1000.

plots:-pointplotseqn,Totientn,n=2..1000,labels=n,`φ`n,color=Niagara BlueGreen,symbol=circle

Compatibility

• 

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

• 

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

See Also

NumberTheory

NumberTheory[InverseTotient]