Generators/primitive roots/primitive elements of Prime fields

Question

Algebra by Michael Artin Exer 3.1.7 (Actually Exer 7 of Ch3.2)

By finding primitive elements, verify that the multiplicative group $\mathbb F_p^{\times}$ is cyclic for all primes $p < 20$.

After some highly tedious manual computation (or excel/sheets, wolfram etc), we obtain that for each prime less than 20, we have, resp, the following as possible generators $1,2,3,3,7,11,11,13$. There are other generators for most of the primes less than 20, but are there patterns or properties possible generators? Like every 2p-5 is a generator of $\mathbb F_p^{\times}$ or every generator of $\mathbb F_p^{\times}$ is odd or prime if p > 5 or something.

Answer 1

My paraphrasing based on this section from Wiki as pointed out as by lhf in h comment to h answer.

For a prime $p$, if the multiplicative order of a number m modulo p is equal to $\varphi \left(p\right) = p-1$ (the order of $\mathbb F_p^{\times}$), then it is a primitive root. In fact the converse is true: If m is a primitive root modulo p, where $p$ is prime, then the multiplicative order of m is $\varphi \left(p\right)=p-1$. We can use this to test for primitive roots.

Answer 2

You have said there are "other generators for most primes". If by most you mean "other the primes 2 and 3" it is ok. Actually from $p\geq5$ one is guaranteed to have more than one generator. The number is given by $\phi(p-1)$.

(The calculation for $p<20$ can not be termed tedious. As one can always omit squares from consideration and as there are just 8 primes up to 20 one can get them all without software in less than 5 minutes.)