Primitive Roots Modulo $2^n$ for $n\geq3$

Question

Question:

(a) Prove that there is no primitive root modulo $2^n$ for any $n\geq3$, where $\bar{a}\in(\mathbb{Z}/2^n\mathbb{Z})^\ast$ is a primitive root modulo $2^n$ if the order of $\bar{a}$ is $\varphi(2^n)$ ($\varphi$ denotes the Euler phi-function).

(b) Prove that $(\mathbb{Z}/2^n\mathbb{Z})^\ast$ is generated by $5$ and $-1$, where $(\mathbb{Z}/2^n\mathbb{Z})^\ast$ is the multiplicative group of congruence classes $\bar{a}$ with $(a,n)=1$.

Motivation: This question is from Stein's Elementary Number Theory: Primes, Congruences, And Secrets (you can find a free legal copy of it here on the author's website), on p. 47, and it was asked in a previous homework of a number theory class I am currently taking.

Part (a) can be proved at least in two ways. The TA for the course gave an inductive argument that shows that $\forall \bar{a}\in(\mathbb{Z}/2^n\mathbb{Z})^\ast:(\bar{a})^{2^{n-2}}=\bar{1}$, from which the result follows immediately since $\varphi(2^n)=2^{n-1}$. This argument goes like this: The statement holds for $n=3$ since any nontrivial element has order $2$. Suppose the statement is true for $n=k\geq3$, that is, $\forall \bar{a}\in(\mathbb{Z}/2^n\mathbb{Z})^\ast: (\bar{a})^{2^{k-2}}=\bar{1}$. Let $a\in\mathbb{Z}$. Then $a^{2^{k-2}}=1+q2^{k}$ for some integer $q$. Squaring both sides gives $a^{2^{k-1}}=1+(q+q^22^{k-1})2^{k+1}$.

Another proof is outlined as a hint in the book, which goes like this (I guess): If for some $n\geq3$ there were a primitive root $\bar{p}$ modulo $2^n$, then $\bar{p}$ would be a generator of $(\mathbb{Z}/2^n\mathbb{Z})^\ast$, as by definition $|(\mathbb{Z}/2^n\mathbb{Z})^\ast|=\varphi(2^n)$. Then $\forall m\in\mathbb{Z},\exists k,l\in\mathbb{Z}:p^k=m+l2^n$, which, reducing modulo $2^3$, gives a contradiction as $(\mathbb{Z}/8\mathbb{Z})^\ast$ has no generators.

I am aware that this question has been asked around one year ago (see here), but I could not make use of the hints provided back then. Also I have a relatively good background in undergraduate algebra, so group theoretic proofs are welcomed as well (for the both parts if you can come up with a different proof for (a)). Finally I wanted to ask about the significance of this question. In particular what is the importance of $(\mathbb{Z}/2^n\mathbb{Z})^\ast$?

Answer 1

A simple proof for seeing there is no primitive root $\bmod 2^n$ is that the powers of $x$ are congruent to $1$ or $x\bmod 8$, so you are missing at least half of the elements.

Now lets prove all elements $\bmod 2^n$ are generated by $5$ and $-1$. There are $2^{n-1}$ congruence classes relatively prime to $2^n \bmod 2^n$. We will prove that the order of $5$ is $2^{n-2}$, if we do this we will have proved $2^ {n-2}$ generates all of the congruence classes that are $1$ or $5\bmod 8$.

The order of $5$ is the smallest positive integer $j$ so that $5^j\equiv 1 \bmod 2^n$ , in other words the smallest power $n$ so that $2^n|5^j-1$ . By the lifting the exponent lemma the largest power of $2$ dividing $2^k-1$ is the largest power of $2$ dividing $k$ plus the largest exponent of $2$ dividing $5-1=4$. Therefore the maximum exponent of $2$ dividing $2^k-2$ is the maximum exponent of $2$ dividing $k$ plus $2$. This tells us the order of $5$ is $2^{n-2}$, since if we want $2^j-1$ to be a multiple of $2^n$, $j$ must be a multiple of $2^{n-2}$.

Then the order of $5$ is $2^{n-2}$ so we can generate all of the residue classes congruent to $1$ or $5$. From here we can generate the other classes by multiplying by $-1$.