# Euler's totient function

## English

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### Etymology

Named after the 18th century Swiss mathematician Leonhard Euler (1707–1783).

### Noun

Euler's totient function ‎(uncountable)

1. (number theory) The function that counts how many integers below a given integer are coprime to it.
Due to Euler's theorem, if f is a positive integer which is coprime to 10, then
${\displaystyle 10^{\phi (f)}\equiv 1{\pmod {f}}}$
where ${\displaystyle \phi }$ is Euler's totient function. Thus ${\displaystyle f|(10^{\phi (f)}-1)}$, which fact which may be used to prove that any rational number whose expression in decimal is not finite can be expressed as a repeating decimal. (To do this, start by splitting the denominator into two factors: one which factors out exclusively into twos and fives, and another one which is coprime to 10. Secondly, multiply both numerator and denominator by such a natural number as will turn the first said factor into a power of 10 (call it N). Thirdly, multiply both numerator and denominator by such a number as will turn the second said factor into a power of 10 minus one (call it M). Fourthly, resolve the numerator into a sum of the form ${\displaystyle aNM+bM+c}$. Then the repeating decimal has the form ${\displaystyle a.b(c)}$ where b may be padded by zeroes (if necessary) to take up ${\displaystyle \log _{10}N}$ digits, and c may be padded by zeroes (if necessary) to take up ${\displaystyle \lceil \log _{10}M\rceil }$ digits.)

#### Usage notes

• Usually denoted with the Greek letter phi (${\displaystyle \phi }$ or ${\displaystyle \varphi }$).