I am not certain about the meaning of p<>2, but if the order of a is 2t, then a^t==-1 Mod p^k....Because there are only two elements of order 2, and phi(2) =1, which tells us only one element belongs to 2. That is, if x^2==1 Mod P^k, then (x-1)(x+1) == 0 Mod P^k. So in an integral domain, one of the terms under discussion (a^t-1) or (a^t+1) will be equal to zero, Mod p^k.
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