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Euler's Totient

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.

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Euler's Totient -

Tag(s) : Mathematics, Arithmetics

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Euler's Totient

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Euler's Totient Phi Calculator


Display the list of numbers relatively prime with N

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.

Answers to Questions

How to calculate phi(n) (Euler's totient)?

Euler Phi totient calculator computes the value of \( \varphi(n) = \phi(n) = phi(n) \) 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, first find 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 \( 1 \) 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 \( \varphi(n) = n-1 \)

What is Euler's totient for?

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

Example: 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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