The Euler Totient Function Ჶ(n) : Number Theory #The #Euler #Totient #Function #Ჶ(n) #Number #Theory

March 04, 2026

The Euler Totient Function Ჶ(n) :

If n ≥ 1 , the Euler Totient Function Ჶ(n) is defined to be the number of positive integers not exceeding n which are relatively prime to n ; thus ,

Ჶ ( n ) = 𝚺 1 for 1 ≤ k ≤ n, here 𝚺 indicates that the sum is extended over those k relatively prime to n.

For Example :

Suppose n = 1.

Since 1 has relatively prime to itself is 1 , Ჶ ( 1 ) = 1.

Suppose n = 2.

Since 2 has positive integer less than 2 which is relatively prime to 2 is 1, Ჶ ( 2 ) = 1.

Suppose n = 3.

Since 3 has positive integers less than 3 which are relatively prime to 3 are 1,2 , Ჶ ( 3 ) = 2.

Suppose n = 4.

Since 4 has positive integers less than 4 which are relatively prime to 4 are 1,3 , Ჶ ( 4 ) = 2.

Suppose n = 5.

Since 5 has positive integers less than 4 which are relatively prime to 4 are 1,2,3,4 , Ჶ ( 5 ) = 4.

Suppose n =6.

Since 6 has positive integers less than 6 which are relatively prime to 6 are 1,5 , Ჶ ( 6 ) = 2.

Here is a short table of values of Ჶ ( n ).

n : 1 2 3 4 5 6 7 8 9 10

Ჶ( n ) : 1 1 2 2 4 2 6 4 6 4

Result :

If n ≥ 1 , we have 𝚺 Ჶ ( d ) = n for d/n

Note :

The sum for Ჶ ( n ) in the above Result can also be expressed as a product extended over the distinct prime divisors of n.

A Product formula for Ჶ ( n ) :

For n ≥ 1 , we have Ჶ ( n ) = n 𝚷 ( 1 - 1/p ).

Properties of Euler's Totient Function :

Ჶ ( Pⁿ ) = Pⁿ - p^n-1 for prime p and n ≥ 1 .
Ჶ ( mn ) = Ჶ ( m ) Ჶ ( n ) ( d/ Ჶ ( d ) ), where d = ( m, n ).
Ჶ ( mn ) = Ჶ ( m ) Ჶ ( n ) if ( m , n ) = 1.
a / b implies Ჶ ( a ) / Ჶ ( b ) .
Ჶ ( n ) is even for n ≥ 3 . Moreover , if n has r distinct odd prime factors , then 2^r / Ჶ ( n ) .