Brocard's problem

Statement

Brocard's problem asks for all pairs of positive integers $(n,m)$ such that:

$n!+1=m^{2}$.

The only known solutions are $(4,5),(5,11),(7,71)$. The $n$s that occur as part of such solution pairs are termed Brown numbers. It is conjectured that there are no other solutions.

Related facts

• abc conjecture implies that the number of solutions to this equation is finite. Further information: anc conjecture implies finitely many Brown numbers