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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!

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# Question about congruences and orders

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