Let $p \ne 2$ be a prime and $a$ the smallest positive integer that is a primitive root modulo $p$. Is $a$ necessarily a primitive root modulo $p^2$ (and hence modulo all powers of $p$)? I checked this for all $p < 3 \times 10^5$ and it seems to work, but I can't see any sound theoretical reason why it should be the case. What is there to stop the Teichmuller lifts of the elements of $\mathbb{F}\_p^\times$ being really small?



Is the smallest primitive root modulo p a primitive root modulo p^2?
It is not true in general. See <http://primes.utm.edu/curios/page.php/40487.html> for the example, 5 mod 40487^2.


