Euler's totient function

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English[edit]

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Etymology[edit]

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

Noun[edit]

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
       10^{\phi(f)} \equiv 1 \pmod f
    where \phi is Euler's totient function. Thus  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 a N M + b M + c. Then the repeating decimal has the form a.b(c) where b may be padded by zeroes (if necessary) to take up \log_{10} N digits, and c may be padded by zeroes (if necessary) to take up \lceil \log_{10} M\rceil digits.)

Usage notes[edit]

  • Usually denoted with the Greek letter phi (\phi or \varphi).

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