1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Urgent: Number Theory-Wilson's Theorem

  1. Nov 2, 2008 #1

    1. let [tex]p[/tex] be prime form [tex]4k+3[/tex] and let [tex]a[/tex] be an integer. Prove that a has order [tex]p-1[/tex] in the group [tex]U(\frac{\texbb{Z}}{p\texbb{Z}})[/tex] iff [tex]-a[/tex] has order [tex]\frac{(p-1)}{2}[/tex]

    2. let [tex]p[/tex] be odd prime explain why: [tex]2*4*...*(p-1)\equiv (2-p)(4-p)*...*(p-1-p)\equiv(-1)^{(p-1)/2}*1*3*...*(p-2) [/tex]mod p.

    3. Using number 2 and wilson's thereom [[tex](p-1)!\equiv-1[/tex] mod p] prove [tex]1^23^25^2*....*(p-2)^2\equiv(-1)^{(p-1)/2}[/tex] mod p

    Last edited: Nov 3, 2008
  2. jcsd
  3. Nov 2, 2008 #2
    For p=3+4k, then (p-1)/2 is an odd number. Thus [tex]1\equiv (-a)^\frac{p-1}{2} =-a^\frac{p-1}{2} [/tex]

    Consequently the right side would be -1 under the circumstances that both powers of a equalled 1.
    Last edited: Nov 2, 2008
  4. Nov 3, 2008 #3
    I got the rest, I still need help in #2.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Urgent: Number Theory-Wilson's Theorem
  1. Wilson's Theorem (Replies: 6)

  2. Wilson's Theorem (Replies: 4)

  3. Number theory theorems (Replies: 17)