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).

Berndt, B. C. and Galway, W. F. "On the Brocard-Ramanujan Diophantine Equation ."
Submitted. http://www.math.uiuc.edu/~galway/Submissions/Ramanujan469.ps
and http://www.math.uiuc.edu/~berndt/articles/galway.pdf.Brocard,
H. Question 166. Nouv. Corres. Math.2, 287, 1876.Brocard,
H. Question 1532. Nouv. Ann. Math.4, 391, 1885.Dabrowski,
A. "On the Diophantine Equation ." Nieuw Arch. Wisk.14, 321-324,
1996.Erdős, P. and Obláth, R. "Über diophantische
Gleichungen der Form
und ."
Acta Szeged8, 241-255, 1937.Gupta, H. "On a Brocard-Ramanujan
Problem." Math. Student3, 71, 1935.Guy, R. K.
"Equations Involving Factorial ." §D25 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 193-194,
1994.Ramanujan, S. Collected
Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar,
and B. M. Wilson). Providence, RI: Amer. Math. Soc., p. 327, 2000.Overholt,
M. "The Diophantine Equation ." Bull. London Math. Soc.25, 104,
1993.Sloane, N. J. A. Sequence A038202
in "The On-Line Encyclopedia of Integer Sequences."Wells,
D. The
Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England:
Penguin Books, pp. 57 and 70, 1986.