The Brocard–Ramanujan problem, which is an unsolved problem in number theory, is to find integer solutions
(
x
,
ℓ
)
(x,\ell )
of
x
2
−
1
=
ℓ
!
x^2-1=\ell !
. Many analogs of this problem are currently being considered. As one example, it is known that there are at most only finitely many algebraic integer solutions
(
x
,
ℓ
)
(x, \ell )
, up to a unit factor, to the equations
N
K
(
x
)
=
ℓ
!
N_K(x) = \ell !
, where
N
K
N_K
are the norms of number fields
K
/
Q
K/\mathbf Q
. In this paper, we construct infinitely many number fields
K
K
such that
N
K
(
x
)
=
ℓ
!
N_K(x) = \ell !
has at least
22
22
solutions for positive integers
ℓ
\ell
.