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Euler's totient function

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EnglishEdit

EtymologyEdit

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

Proper nounEdit

Euler's totient function

  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
        
    where   is Euler's totient function. Thus  , 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  . Then the repeating decimal has the form   where b may be padded by zeroes (if necessary) to take up   digits, and c may be padded by zeroes (if necessary) to take up   digits.)

Usage notesEdit

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

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