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Section 9.2 The Euler Phi Function

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We give the size of the group of units (mod n) a special name.

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Definition 9.2.1.

We give the order of Un the name Ο•(n). That is, by definition,

Ο•(n)=|Un|.

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This is the so-called Euler Ο• function. It can also be written phi, it is pronounced β€˜fee’, and it's occasionally notated Ο† just for fun. We'll meet Euler many times in this text; see Historical remark 13.0.3.

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Remark 9.2.2.

Since modulo one everything is one, we say U1={[1]} and Ο•(1)=1 since gcd(1,1)=1, despite the fact that also everything is zero. If this bothers you, you are nearly at the algebraic notion of a field mentioned toward the end of Section 8.1, and may wish to read some discussions of the field with one element.

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One of the most fun things to do with basic number theory is to explore new concepts with pencil and paper – because it really is tractable.

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Question 9.2.3.

Do you see any patterns on the value of Ο•(n)?

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Subsection 9.2.1 Euler's theorem

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So far this is a relatively abstract concept. What follows is not abstract at all, but very, very useful! Let's follow the following argument to see what we can find out about Ο•(n).

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Recall the notion of the order of an element (Definition 8.3.10). So any random element [a]∈Un (for some n) has an order.

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Example 9.2.4.

For instance, the order of [2] in U7 is 3, because [2]1 and [2]2 are not 1, but [2]3≑8≑1 (mod 7).

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This means we can apply the things we learned about orders, in particular Theorem 8.3.12 of Lagrange. It stated that the order of any element of a finite group divides the order of the group itself.

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Think about what this implies for orders in |Un|. First, |a| divides |Un|. (For instance, in Example 9.2.4, 3 divides 6.) That can be rewritten as

|a|βˆ£Ο•(n), or Ο•(n)=k|a|

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for some positive integer k.

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Finally, let's apply this fact to powers of a.

aΟ•(n)=ak|a|=(a|a|)k≑1k≑1 (mod n)

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This is very interesting; without it, all we would know is that a|a|≑1 because that's the definition of what β€˜order’ means. With it, we have proved one of the many celebrated theorems of Leonhard Euler:

See the preceding paragraphs.

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Try verifying Euler's Theorem for n=12 and n=11 for some simple a such as a=3 or a=5. Can you see how to recover Fermat's Little Theorem from Euler's Theorem, as a special case? (See Exercise 9.6.2.)