Brocard's problem asks to find the values of 
for which 
is a square number 
, where 
is the factorial (Brocard 1876,
1885). The only known solutions are 
, 5, and 7. Pairs of numbers 
are called Brown numbers.
In 1906, Gérardin claimed that, if 
, then 
must have at least 20 digits. Unaware of Brocard's query,
Ramanujan considered the same problem in 1913. Gupta (1935) stated that calculations
of 
up to 
gave no further solutions.
It is virtually certain that there are no more solutions (Guy 1994). In fact, Dabrowski (1996) has shown that 
has only finitely many solutions for general 
, although this result requires assumption of a weak form of
the abc conjecture if 
is square).
There are no other solutions with 
(Wells 1986, p. 70), and Berndt and Galway have
further searched up to 
without finding any further solutions.
Wilson has also computed the least 
such that 
is square starting at 
, giving 1, 1, 3, 1, 9, 27, 15, 18, 288, 288, 420, 464, 1856,
... (OEIS A038202).
."
Submitted. 
." Nieuw Arch. Wisk. 14, 321-324,
1996.Erdős, P. and Obláth, R. "Über diophantische
Gleichungen der Form 
und 
."
Acta Szeged 8, 241-255, 1937.Gupta, H. "On a Brocard-Ramanujan
Problem." Math. Student 3, 71, 1935.Guy, R. K.
"Equations Involving Factorial 
." §D25 in 
." Bull. London Math. Soc. 25, 104,
1993.Sloane, N. J. A. Sequence