# eulersPhi: Eulers's Phi Function In numbers: Number-Theoretic Functions

## Description

Euler's Phi function (aka Euler's ‘totient’ function).

## Usage

 `1` ```eulersPhi(n) ```

## Arguments

 `n` Positive integer.

## Details

The `phi` function is defined to be the number of positive integers less than or equal to `n` that are coprime to `n`, i.e. have no common factors other than 1.

## Value

Natural number, the number of coprime integers `<= n`.

## Note

Works well up to `10^9`.

`primeFactors`, `Sigma`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```eulersPhi(9973) == 9973 - 1 # for prime numbers eulersPhi(3^10) == 3^9 * (3 - 1) # for prime powers eulersPhi(12*35) == eulersPhi(12) * eulersPhi(35) # TRUE if coprime ## Not run: x <- 1:100; y <- sapply(x, eulersPhi) plot(1:100, y, type="l", col="blue", xlab="n", ylab="phi(n)", main="Euler's totient function") points(1:100, y, col="blue", pch=20) grid() ## End(Not run) ```

### Example output

```[1] TRUE
[1] TRUE
[1] TRUE
```

numbers documentation built on May 15, 2021, 1:08 a.m.