Brocard's problem

Not to be confused with Brocard's conjecture.

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.

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

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,^[1][2] and independently in 1913 by Srinivasa Ramanujan.^[3]

Brown numbers

Pairs of the numbers ${\displaystyle (n,m)}$ that solve Brocard's problem were named Brown numbers by Clifford A. Pickover in his 1995 book Keys to Infinity, after learning of the problem from Kevin S. Brown.^[4] As of October 2022, there are only three known pairs of Brown numbers:

(4,5), (5,11), and (7,71),

based on the equalities

4! + 1 = 5² = 25,

5! + 1 = 11² = 121, and

7! + 1 = 71² = 5041.

Paul Erdős conjectured that no other solutions exist.^[5] Computational searches up to one quadrillion have found no further solutions.^[6][7][8]

Connection to the abc conjecture

It would follow from the abc conjecture that there are only finitely many Brown numbers.^[9] More generally, it would also follow from the abc conjecture that $${\displaystyle n!+A=k^{2}}$$ has only finitely many solutions, for any given integer ${\displaystyle A}$,^[10] and that $${\displaystyle n!=P(x)}$$ has only finitely many integer solutions, for any given polynomial ${\displaystyle P(x)}$ of degree at least 2 with integer coefficients.^[11]