Tool to compute Phi: Euler Totient. Euler's Totient φ(n) is a number representing the number of integers inferior to n, relatively prime with n.

Euler's Totient - dCode

Tag(s) : Mathematics,Arithmetics

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Tool to compute Phi: Euler Totient. Euler's Totient φ(n) is a number representing the number of integers inferior to n, relatively prime with n.

The value of \( \varphi(n) = \phi(n) = phi(n) \) is computed in several ways, the best known formula is $$ \varphi(n) = n \prod_{p \mid n} \left( 1 - \frac{1}{p} \right) $$

where \( p \) is a prime factor which divides \( n \).

To calculate the value of the Euler indicator, you must know the prime factor decomposition of \( n \). Consider \( p_i \) the \( m \) distinct prime factors of \( n \), then the formula becomes:

$$ \varphi(n) = n \prod_{i=1}^m \left( 1 - \frac{1}{p_i} \right) $$

For \( n = 6 \), only the numbers \( 4 \) and \( 5 \) are coprime with \( 6 \) so \( \varphi(6) = 2 \). This is confirmed by the formula for \( n = 6 = 2^1 \times 3^1 \) you get: $$ \varphi(6) = 6 (1-\frac{1}{2}) (1-\frac{1}{3}) = 2 $$

If \( n \) is a prime number, then \( \varphi(n) = n-1 \)

Euler totient function is used in modular arithmetic. It is used in Euler's theorem :

If \( n \) is an integer superior or equal to 1 and \( a \) an integer coprime with \( n \), then $$ a^{\phi(n)} \equiv 1 \mod n $$

n=7, a=3 and phi(7) = 6 so 3^6 = 729 = 1 modulo 7

This theorem is the basis of the RSA encryption.

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