We estimate character sums with
n
!
n!
, on average, and individually. These bounds are used to derive new results about various congruences modulo a prime
p
p
and obtain new information about the spacings between quadratic nonresidues modulo
p
p
. In particular, we show that there exists a positive integer
n
≪
p
1
/
2
+
ε
n\ll p^{1/2+\varepsilon }
such that
n
!
n!
is a primitive root modulo
p
p
. We also show that every nonzero congruence class
a
≢
0
(
mod
p
)
a \not \equiv 0 \pmod p
can be represented as a product of 7 factorials,
a
≡
n
1
!
…
n
7
!
(
mod
p
)
a \equiv n_1! \ldots n_7! \pmod p
, where
max
{
n
i
|
i
=
1
,
…
,
7
}
=
O
(
p
11
/
12
+
ε
)
\max \{n_i \ |\ i=1,\ldots , 7\}=O(p^{11/12+\varepsilon })
, and we find the asymptotic formula for the number of such representations. Finally, we show that products of 4 factorials
n
1
!
n
2
!
n
3
!
n
4
!
,
n_1!n_2!n_3!n_4!,
with
max
{
n
1
,
n
2
,
n
3
,
n
4
}
=
O
(
p
6
/
7
+
ε
)
\max \{n_1, n_2, n_3, n_4\}=O(p^{6/7+\varepsilon })
represent “almost all” residue classes modulo p, and that products of 3 factorials
n
1
!
n
2
!
n
3
!
n_1!n_2!n_3!
with
max
{
n
1
,
n
2
,
n
3
}
=
O
(
p
5
/
6
+
ε
)
\max \{n_1, n_2, n_3\}=O(p^{5/6+\varepsilon })
are uniformly distributed modulo
p
p
.