Brocard's problem asks to find the positive integers 
for which 
is a square number 
with a positive
integer 
,
where 
is the factorial (Brocard 1876, 1885). The only known
solutions are 
,
5, and 7. Pairs of numbers 
are called Brown numbers.
Gérardin (1906) claimed that, if 
, then 
must have at least 20 digits. Unaware of Brocard's query,
Ramanujan (1913) considered the same problem. Gupta (1935) stated that calculations
of 
up to 
gave no further solutions.
It is conjectured that there are no more solutions (Guy 1994). Dabrowski (1996) proved that, for each fixed positive integer 
that is not a square number,
the Diophantine equation 
has only finitely many solutions in positive
integers 
and 
.
For square 
, his finiteness result assumes a weak consequence of the abc conjecture (Berndt and Galway 2000).
There are no other solutions with 
(Wells 1986, p. 70), and Berndt and Galway have
further searched up to 
without finding any further solutions (Berndt and Galway
2000). Epstein and Glickman (2020) reported no further solutions for 
.
Wilson has also computed the least positive integer 
such that 
is a square number starting
at 
,
giving 1, 1, 3, 1, 9, 27, 15, 18, 288, 288, 420, 464, 1856, ... (OEIS A038202).
."
Ramanujan J. 4, 41-42, 2000. 
." Nieuw Arch. Wisk. 14, 321-324,
1996.Epstein, A. and Glickman, J. "brocard." GitHub, 2020.
und 
." Acta Szeged 8, 241-255, 1937.Gérardin,
A. "Contribution à l'étude de l'équation 
." Nouv. Ann. Math. 6, 222-226,
1906. 
." §D25 in 
." Bull. London Math. Soc. 25, 104,
1993.Ramanujan, S. Question 469. J. Indian Math. Soc. 5,
59, 1913.Ramanujan, S.