## Generators of multiplicative group $$(\mathbb{Z}/2^{\alpha}\mathbb{Z})^*$$

The set of all unit elements in the cyclic group $$\mathbb{Z}/N\mathbb{Z}$$ is a group under multiplication modulo $$N$$ of order $$\phi(N)$$, where $$\phi$$ is the Euler-totient function. It is denoted by $$(\mathbb{Z}/N\mathbb{Z})^*$$.
It is well known that $$(\mathbb{Z}/N\mathbb{Z})^*$$ is not always cyclic (Primitive element theorem) (See Chater 10, [1]). For example $$(\mathbb{Z}/8\mathbb{Z})^*=\{1,3,5,7\}$$ is not cyclic. $$(\mathbb{Z}/2\mathbb{Z})^*=\{1\}$$ and $$(\mathbb{Z}/4\mathbb{Z})^*=\{1,3\}$$ are trivially cyclic.

Question: Show that for $$\alpha\geq 3$$, $$(\mathbb{Z}/2^{\alpha}\mathbb{Z})^*$$ is generated by $$-1$$ and 5?
You may use the following sage code for validating the result:
Reference: [1] T M Apostol, Introduction to Analytics Number Theory