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Euler's Totient

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.

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Euler's Totient -

Tag(s) : Arithmetics

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# Euler's Totient

## Solver for Phi(?)=N (Inverse Phi)

### What is Euler's totient? (Definition)

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 $n$

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

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 $\varphi(n) = n-1$

### How to calculate inverse phi(n)?

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)

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

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.

### What are Euler's totient properties?

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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