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Brocard's Problem

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, and there are no other solutions $<1027$. The pairs of numbers $(m,n)$ are called Brown Numbers.

See also Brown Numbers, Factorial, Square Number


Brocard, H. Question 166. Nouv. Corres. Math. 2, 287, 1876.

Brocard, H. Question 1532. Nouv. Ann. Math. 4, 391, 1885.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 193, 1994.

© 1996-9 Eric W. Weisstein