Waste less time on Facebook — follow Brilliant.
×

Divisibility of Euler's Totient Function

Let \(p\) be a prime number and \(n\) be a positive integer. Prove that \(\phi(p^n-1)\) is divisible by \(n\), where \(\phi\) denotes Euler's totient function.

Note by Finn Hulse
3 years, 7 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.
2 \times 3 \( 2 \times 3 \)
2^{34} \( 2^{34} \)
a_{i-1} \( a_{i-1} \)
\frac{2}{3} \( \frac{2}{3} \)
\sqrt{2} \( \sqrt{2} \)
\sum_{i=1}^3 \( \sum_{i=1}^3 \)
\sin \theta \( \sin \theta \)
\boxed{123} \( \boxed{123} \)

Comments

Sort by:

Top Newest

Firstly,

\(p^n\equiv1\mod(p^n-1)\) Note that n is the smallest number with that property, in other words n\(=ord_{p^n-1}(p)\)

Also,

\(p^{\varphi(p^n-1)}\equiv1\mod(p^n-1)\)

But the order must divide every number with that property , so

\(n|\varphi(p^n-1)\)

Bogdan Simeonov - 3 years, 7 months ago

Log in to reply

Beautiful. :D

Finn Hulse - 3 years, 7 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...