Tool to compute Phi: Euler Totient. Euler's Totient φ(n) represents the number of integers inferior to n, coprime with n.

Euler's Totient - dCode

Tag(s) : Arithmetics

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Tool to compute Phi: Euler Totient. Euler's Totient φ(n) represents the number of integers inferior to n, coprime with n.

Euler's totient, noted with the greek letter phi : \( \varphi(n) \) or \( \phi(n) \) is the value representing the number of integers less than \( n \) that are coprime with \( n \)

Euler Phi totient calculator computes the value of Phi(n) 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/totient, first find the prime factor decomposition of \( n \). If \( p_i \) are 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 \( 1 \) and \( 5 \) are coprime with \( 6 \) so \( \varphi(6) = 2 \). This is confirmed by the formula for \( n = 6 = 2^1 \times 3^1 \), as: $$ \varphi(6) = 6 (1-\frac{1}{2}) (1-\frac{1}{3}) = 2 $$

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

Euler totient phi 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 $$ a^{\varphi(n)} \equiv 1 \mod n $$

Example: \( n=7 \) , \( a=3 \) and \( \varphi(7) = 6 \) so \( 3^6 = 729 \equiv 1 \mod 7 \)

This theorem is the basis of the RSA encryption.

The Euler indicator is an essential function of modular arithmetic:

- A positive integer \( p \) is a prime number if and only if \( \varphi(p) = p - 1 \)

- The value \( \varphi(n) \) is even for all \( n > 2 \)

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