Results 1 to 3 of 3

Math Help - Proof question

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    19

    Proof question

    Recall the definition of n! (read n factorial"):
    n! = (n)(n-1)(n-2) .(2)(1) =∏(k)
    In both (a) and (b) below, suppose p≥3 is prime.
    (a) Prove that if x∈ Zpx is a solution to x square ≡1 (mod p), then x ≡1 (mod p).
    (b) Prove that (p-1)!≡1 (mod p)

    Zpx x shoud be above p
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member Haven's Avatar
    Joined
    Jul 2009
    Posts
    197
    Thanks
    8
    Okay, Correct me if I'm wrong, but the two question are
    Let p be prime.
    1) If x\in\mathbb{Z}_{p} and  x^{2}\equiv\\1 (mod p) then  x\equiv\pm\\1 (mod p)
    2)  (p - 1)!\equiv\pm\\1 (mod p)

    For 1 we get
     x^{2}\equiv\\1 (mod p)
     \Rightarrow\\x^{2} -1\equiv\\0 (mod p)
     \Rightarrow\\(x-1)(x+1)\equiv\\0 (mod p)
     \Rightarrow\\p|(x-1)(x+1)
     \Rightarrow\\p|(x-1) or  p|(x+1)
     \Rightarrow\\x\equiv\\1 (mod p) or  x\equiv\\-1 (mod p)
     \Rightarrow\\x\equiv\pm\\1 (mod p)

    2) is wrong. The correct version is  (p - 1)!\equiv\\1 (mod p), the result is known as Wilson's Theorem
     (p-1)! = 1*2*3........(p-2)(p-1)
    Now 1 and p-1 are the only numbers modulo p that are their own inverses.
    For any other number x such that 1<x<p-1 there exists x^{-1} such that 1<x^{-1}<p-1 and x*x^{-1}\equiv\\1 (mod p).
    So we match up each x with its inverse and we get:
     (p-1)!\equiv\\1*2*3........(p-2)(p-1)\equiv\\1*1*1.......(p-1)\equiv\\p-1\equiv\\-1 (mod p)
    Now we have the desired result.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Sep 2009
    Posts
    19
    Thank you for your answer

    but for the second one

    it is Wilson's theorem like you said

    but the right one should be (p-1)!≡-1
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Proof Question 1
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 29th 2009, 07:47 AM
  2. Another proof question
    Posted in the Calculus Forum
    Replies: 1
    Last Post: December 21st 2008, 11:19 PM
  3. Proof question
    Posted in the Trigonometry Forum
    Replies: 1
    Last Post: November 5th 2008, 05:31 PM
  4. [SOLVED] Proof Question
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: August 29th 2006, 10:56 AM
  5. Proof Question
    Posted in the Algebra Forum
    Replies: 2
    Last Post: March 19th 2006, 06:54 AM

Search Tags


/mathhelpforum @mathhelpforum