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

Euler's Totient - dCode

Tag(s) : Arithmetics

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**Euler's totient** function (or Euler's indicator), 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, the first step is to find the prime factor decomposition of $ n $. If $ p_i $ are the $ m $ distinct prime factors divisors 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 $

Solving $ \phi(x) = N $ requires an optimized search algorithm based on $ \phi(x) \geq \sqrt{\frac{x}{2}} $ and test all values. More details here (link)

**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 $

- $ \varphi(ab) = \varphi(a) \varphi(b) \frac{d}{\varphi(d)} $ with $ d $ the GCD of $ a $ and $ b $

- If $ a $ and $ b $ are coprimes (relatively primes), then $ \varphi(a \times b) = \varphi(a) \times \varphi(b) $

- If $ a $ divides $ b $ then $ \varphi(a) \mid \varphi(b) $

- If $ a $ is even, $ \varphi(2a) = 2 \varphi(a) $

- If $ a $ is odd, $ \varphi(2a) = \varphi(a) $

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