Brocard's problem

Unsolved problem in mathematics

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

More unsolved problems in mathematics

Brocard's problem is a problem in mathematics that seeks integer values of $n$ such that $n!+1$ is a perfect square, where $n!$ is the factorial. Only three values of $n$ are known — 4, 5, 7 — and it is not known whether there are any more.

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

Brown numbers

Pairs of the numbers $(n,m)$ 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.Template:R As of October 2022, there are only three known pairs of Brown numbers: Template:Bi based on the equalities Template:Bi Template:Bi Template:Bi Paul Erdős conjectured that no other solutions exist.Template:R Computational searches up to one quadrillion have found no further solutions.Template:R

Connection to the abc conjecture

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

External links

• Template:Mathworld2