Let
g
(
p
)
g(p)
denote the least primitive root modulo
p
p
, and
h
(
p
)
h(p)
the least primitive root modulo
p
2
p^2
. We computed
g
(
p
)
g(p)
and
h
(
p
)
h(p)
for all primes
p
≤
10
16
p\le 10^{16}
. As a consequence we are able to prove that
g
(
p
)
>
p
5
/
8
g(p)>p^{5/8}
for all primes
p
>
3
p>3
and that
h
(
p
)
>
p
2
/
3
h(p)>p^{2/3}
for all primes
p
p
. More generally, we provide values of
p
α
p_\alpha
where
g
(
p
)
>
p
α
g(p)>p^\alpha
when
p
>
p
α
p>p_\alpha
, for various values of
α
\alpha
with
1
/
2
>
α
>
5
/
8
1/2>\alpha >5/8
. Additionally, we give a log-histogram of
g
(
p
)
g(p)
when
g
(
p
)
≥
100
g(p)\ge 100
and empirical evidence that
g
(
p
)
≪
(
log
p
)
(
log
log
p
)
2
g(p)\ll (\log p)(\log \log p)^2
.