Group Theory: Showing the Order of Element 2 is 2k

[lor]

Homework Statement

Let p=2^(k)+1 , in which k is a positive integer, be a prime number. Let G be the group of integers 1, 2, ... , p-1 under multiplication defined modulo p.

By first considering the elements 2^1, 2^2, ... , 2^k and then the elements 2^(k+1), 2^(k+2), ... show that the order of the element 2 is 2k.

Deduce that k= 2^n for n is a natural number.

Homework Equations

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The Attempt at a Solution

My problem is that I don't understand the question, or how to get started. I'm looking for some hints on how to approach this. I figured that the remainder of (2^k)/(p^(k)+1) should be zero for k to be the order. Is this right? I really have no idea what I'm supposed to do ;/


This is my first time posting on here so I hope I've set this out right! Thanks in advance :)

 

[praharmitra]

For the first part, you are to show that order of the element 2 is 2k, which means
Show -
2^{2k} (\tex{mod} p) \equiv 1