Want to win a PS4? Go Premium and enter to win our High-Tech Treats giveaway. Enter to Win

x
?
Solved

Euler's totient/phi function

Posted on 2006-06-27
5
Medium Priority
?
900 Views
Last Modified: 2012-05-05
If gcd(a,n) = 1 and gcd(a-1,n) = 1 then prove that

1 + a + a^2 + a^3 + ... + a^(phi-1) = 0 (mod n)

Where phi = phi(n) is the Euler phi/totient function.

= means "is congruent to"

I can prove this if n is a prime number, but I need the proof for any n > 1.

a and n are integers, of course.
0
Comment
Question by:acerola
[X]
Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

  • Help others & share knowledge
  • Earn cash & points
  • Learn & ask questions
  • 2
5 Comments
 
LVL 5

Expert Comment

by:bastibartel
ID: 17044565
Hi there ?

I don't exactly undertand your notation:
1 + a + a^2 + a^3 + ... + a^(phi-1) = 0 (mod n)

the part w/o (mod n) does not depend on n

Secondly, can you prove it for n=1 ?

Cheers,
Sebastian
0
 
LVL 1

Author Comment

by:acerola
ID: 17046672
phi = phi(n) (phi is a function of n)

For n=1 it is trivial, since all numbers are congruent to zero modulus 1.

I have already solved it. The sum is a geometric series. The first element is 1 and the scale facor is a, so:

a^0 + a^1 + a^2 + a^3 + ... + a^(phi-1) = (a^(phi) - 1)/(a - 1)

We know that:

a^(phi(n)) = 1 (mod n)

So:

a^(phi) - 1 = 0 (mod n)
since gcd(a-1,n) = 1, we can divide both sides by (a-1)
(a^(phi) - 1)/(a-1) = 0 (mod n)

That's it. I didn't realize that it was a geometric series. I was trying to solve it using Newton's binomial...
0
 
LVL 5

Expert Comment

by:bastibartel
ID: 17046716
*closed* :-)
0
 
LVL 5

Accepted Solution

by:
Netminder earned 0 total points
ID: 17076348
Closed, 250 points refunded.
Netminder
Site Admin
0

Featured Post

Independent Software Vendors: We Want Your Opinion

We value your feedback.

Take our survey and automatically be enter to win anyone of the following:
Yeti Cooler, Amazon eGift Card, and Movie eGift Card!

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Have you ever thought of installing a power system that generates solar electricity to power your house? Some may say yes, while others may tell me no. But have you noticed that people around you are now considering installing such systems in their …
Article by: Nicole
This is a research brief on the potential colonization of humans on Mars.
Finds all prime numbers in a range requested and places them in a public primes() array. I've demostrated a template size of 30 (2 * 3 * 5) but larger templates can be built such 210  (2 * 3 * 5 * 7) or 2310  (2 * 3 * 5 * 7 * 11). The larger templa…
I've attached the XLSM Excel spreadsheet I used in the video and also text files containing the macros used below. https://filedb.experts-exchange.com/incoming/2017/03_w12/1151775/Permutations.txt https://filedb.experts-exchange.com/incoming/201…
Suggested Courses

618 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.

Join & Ask a Question