Brocard's Problem

Brocard's problem asks to find the values of n for which n!+1 is a square number m^2, where n! is the factorial (Brocard 1876, 1885). The only known solutions are n=4, 5, and 7. Pairs of numbers (m,n) are called Brown numbers. In 1906, Gérardin claimed that, if m>71, then m must have at least 20 digits. Unaware of Brocard's query, Ramanujan considered the same problem in 1913. Gupta (1935) stated that calculations of n! up to n=63 gave no further solutions.

It is virtually certain that there are no more solutions (Guy 1994). In fact, Dabrowski (1996) has shown that n!+A=k^2 has only finitely many solutions for general A, although this result requires assumption of a weak form of the abc conjecture if A is square).

There are no other solutions with n<=10^7 (Wells 1986, p. 70), and Berndt and Galway have further searched up to n=10^9 without finding any further solutions.

Wilson has also computed the least k such that n!+k^2 is square starting at n=4, giving 1, 1, 3, 1, 9, 27, 15, 18, 288, 288, 420, 464, 1856, ... (OEIS A038202).

See also

Brown Numbers, Factorial, Square Number

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Berndt, B. C. and Galway, W. F. "On the Brocard-Ramanujan Diophantine Equation n!+1=m^2." Submitted. and, 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 x!+A=y^2." Nieuw Arch. Wisk. 14, 321-324, 1996.Erdős, P. and Obláth, R. "Über diophantische Gleichungen der Form n!=x^p+/-y^p und n!+/-m!=x^p." Acta Szeged 8, 241-255, 1937.Gupta, H. "On a Brocard-Ramanujan Problem." Math. Student 3, 71, 1935.Guy, R. K. "Equations Involving Factorial n." §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 n!+1=m^2." 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.

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Brocard's Problem

Cite this as:

Weisstein, Eric W. "Brocard's Problem." From MathWorld--A Wolfram Web Resource.

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