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Section 10.7 All the Primitive Roots

There is more to the primitive root story, but we won’t cover the rest in detail. The complete story of which \(n\) have groups of units \(U_n\) that are cyclic is given by Sage. Recall from Sage note 5.3.8 that the question mark gives us information.
Notice that we already showed that bigger powers of two do not have primitive roots, so we have seen parts of both what does and what doesn’t have a primitive root.
To make this result somewhat more plausible, the following cell demonstrates Exercise 10.6.11 – that an odd primitive root for a prime power is also a primitive root for twice that modulus.
This is also consistent with what we already know, since \(\phi(2p^e)=\phi(p^e)\text{.}\) Do the patterns in the interact help you think how you might solve the exercise?
Finally, to really stretch yourself, how do you think you would get from a primitive root modulo \(p\) to one modulo \(p^e\text{?}\) How would you show that other numbers do not have one?