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The many facets of euler’s totient

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Handbook of Number Theory II

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3.1 Introduction

Motivated by a generalization of Fermat’s divisibility theorem, in 1760 L. Euler [133] proved that

((1))

where ϕ(n) denotes the number of positive integers r<n such that (r, n)=1. Euler studied also other important properties of this function, e.g. the multiplicativity property, which means that ϕ(mn)=ϕ(m)ϕ(n) whenever (m, n)=1. (2)

It was C. F. Gauss (see L. E. Dickson [103]) who introduced the symbol ϕ(n) for Euler’s ϕ-function, making the convention ϕ(1)=1. In 1879 J. J. Sylvester introduced the word “totient”, while E. Prouhet (1845) proposed the name “indicator” (see [103]). Sylvester defined also the “totatives” of n to be the integers r<n with (r, n)=1.

Euler’s totient is of major interest in number theory, as well as many other fields of mathematics. For example, the apparently simple result (1) allowed mathematicians to build a code which is almost impossible to break, even though the key is made public (see for instance R. Rivest, A. Shamir and L. Adleman...

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Babeş-Bolyai University of Cluj, Cluj-Napoca, Romania

    J. Sándor

  2. formerly the Technical University of Timişoara, Timişoara, Romania

    B. Crstici

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  1. J. Sándor
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Sándor, J., Crstici, B. (2004). The many facets of euler’s totient. In: Handbook of Number Theory II. Springer, Dordrecht. https://doi.org/10.1007/1-4020-2547-5_3

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  • DOI: https://doi.org/10.1007/1-4020-2547-5_3

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