Results 1 to 3 of 3

Math Help - Congruences, Fermat's Theorem, Wilson's Theorem - General Questions

  1. #1
    Newbie
    Joined
    Oct 2012
    From
    District of Columbia
    Posts
    16

    Congruences, Fermat's Theorem, Wilson's Theorem - General Questions

    Here are several problems that I have been trying to work on but have not gotten very far. Any thoughts?

    1) Suppose that p is an odd prime. Show that 1^(p-1) + 2^(p-1) + ... + (p-1)^(p-1) is congruent to -1 (mod p).

    2) Verify that sigma(p^n) - p^n = ((p^n) - 1)/(p-1) for n = 1,2,...

    3)In assignment 4, we showed that Z_m - {0} = {1,2,...,m-1} is not always a group under multiplication modulo m. Write Z*_m for the set of all elements in Z_m which have a multiplicative inverse in Z_m.
    (a) Prove that a in Z*_m if and only if (a; m) = 1. Conclude that Z*_m has exactly phi(m) elements.
    (b) Verify that Z*_m is a group under multiplication modulo m. Conclude that a phi(m)= 1 for a in Z*_m.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member MaxJasper's Avatar
    Joined
    Aug 2012
    From
    Canada
    Posts
    482
    Thanks
    55

    Lightbulb Re: Congruences, Fermat's Theorem, Wilson's Theorem - General Questions

    1)
    Using Fermat's Little Theorem:

    1^{p-1}+2^{p-1}+\cdots +(p-1)^{p-1}\equiv 1+1+\cdots +1=p-1\equiv -1 (\text{mod} p)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2012
    From
    India
    Posts
    61
    Thanks
    3

    Re: Congruences, Fermat's Theorem, Wilson's Theorem - General Questions

    Quote Originally Posted by ncshields View Post
    Here are several problems that I have been trying to work on but have not gotten very far. Any thoughts?

    1) Suppose that p is an odd prime. Show that 1^(p-1) + 2^(p-1) + ... + (p-1)^(p-1) is congruent to -1 (mod p).

    2) Verify that sigma(p^n) - p^n = ((p^n) - 1)/(p-1) for n = 1,2,...

    3)In assignment 4, we showed that Z_m - {0} = {1,2,...,m-1} is not always a group under multiplication modulo m. Write Z*_m for the set of all elements in Z_m which have a multiplicative inverse in Z_m.
    (a) Prove that a in Z*_m if and only if (a; m) = 1. Conclude that Z*_m has exactly phi(m) elements.
    (b) Verify that Z*_m is a group under multiplication modulo m. Conclude that a phi(m)= 1 for a in Z*_m.
    Your problem 2 looks like a general series summation to me, nothing specific to p being a prime. Please clarify Sigma(p^n) notation here. Is it summation on n, from 0 to say N?

    Salahuddin
    Maths online
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Using Congruences - Wilson's and Fermat's Thms
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: October 26th 2012, 04:27 AM
  2. Replies: 3
    Last Post: June 12th 2012, 07:30 AM
  3. Wilson's theorem proofing questions?
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: May 9th 2012, 04:41 PM
  4. Prove Wilson's theorem by Lagrange's theorem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: April 10th 2010, 02:07 PM
  5. Applying Fermat and Wilson's Theorem
    Posted in the Number Theory Forum
    Replies: 4
    Last Post: February 21st 2010, 07:06 PM

Search Tags


/mathhelpforum @mathhelpforum