Brocard's problem

Unsolved problem in mathematics

Does ${\displaystyle n!+1=m^{2}}$ have integer solutions other than ${\displaystyle n=4,5,7}$?

More unsolved problems in mathematics

Brocard's problem is a problem in mathematics that seeks integer values of ${\displaystyle n}$ such that ${\displaystyle n!+1}$ is a perfect square, where ${\displaystyle n!}$ is the factorial. Only three values of ${\displaystyle n}$ are known — 4, 5, 7 — and it is not known whether there are any more. Though research has extended far beyond n > 7, no additional solutions to the equation n! + 1 = m² are known.

More formally, it seeks pairs of integers ${\displaystyle n}$ and ${\displaystyle m}$ such that$${\displaystyle n!+1=m^{2}.}$$The problem was posed by Henri Brocard in a pair of articles in 1876 and 1885, and independently in 1913 by Srinivasa Ramanujan.