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Question about congruences and orders

  1. Oct 23, 2006 #1
    The question is:

    Show that if p is an odd prime and ord(p^a)a=2t, then
    a^t== -1 mod p^a

    First, I used ord(p^a)a to mean "order of a, mod p^a" and the == sign means congruent.

    So first, I tried a few examples. Let p=3, a=2
    Since ord(9)2=6, then t=3 and:
    2^3 == -1 mod 9 TRUE

    I continued with different values of p and a. Here is a table(sorry it looks weird):

    p--a--t--p^a
    3--2--3--9
    5--2--10--25
    11--2--55--121
    13--2--78--169
    17--2--68--289

    It seems that p|t in all of my examples but I'm stuck....THANKS!
     
  2. jcsd
  3. Oct 24, 2006 #2

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    If we take a^t and square it what do we get? Now, since a^t is not 1, since 2t is the order of a, what do you need to show?
     
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