nt.number theory - Is the smallest primitive root modulo p a primitive root modulo p^2?

The key term here is: Wieferich prime base $a$.

What you observed can be presented to children in the following form: if $p$ is a prime greater than 5 and the fraction $1/p$ has decimal period $d$, numerical tables show $1/p^2$ has decimal period $dp$, $1/p^3$ has decimal period $dp^2$, and generally the decimal period of $1/p^k$ is $dp^{k-1}$. For example, 1/13 has decimal period 6, 1/169 has decimal period $78 = 6 cdot 13$, and 1/2197 has decimal period $1014 = 6 cdot 13^2$.

This works for primes below 100, but if you search far enough you will find a counterexample. The first one is $p = 487$: 1/487 and $1/487^2$ both have decimal period 486. The second counterexample is $p = 56,598,313$. (!!) This list has been Sloaned: http://oeis.org/A045616.

For a general article about this business, see http://www.jstor.org/stable/3219294.

Within algebraic number theory, this phenomenon appears when you compute the ring of integers of ${mathbf Q}(sqrt[n]{2})$, which turns out to be ${mathbf Z}[sqrt[n]{2}]$ for all $n leq 1000$. With that evidence you might guess the ring of integers is always ${mathbf Z}[sqrt[n]{2}]$, just like the ring of integers of ${mathbf Q}(zeta_n)$ is always ${mathbf Z}[zeta_n]$. But in fact it's not always true. There are $n > 1000$ such that ${mathbf Z}[sqrt[n]{2}]$ is not the full ring of integers of ${mathbf Q}(sqrt[n]{2})$. If you search for Wieferich primes to base 2 you will find them.