# Brocard's Problem

## Unsolved Problem

For which pairs of (strictly) positive integers $\tuple {m, n}$ do the following hold:

$n! + 1 = m^2$

The only known pairs are:

 $\text {(1)}: \quad$ $\, \displaystyle \tuple {5, 4}: \,$ $\displaystyle 4! + 1$ $=$ $\displaystyle 24 + 1 = 25 = 5^2$ $\text {(2)}: \quad$ $\, \displaystyle \tuple {11, 5}: \,$ $\displaystyle 5! + 1$ $=$ $\displaystyle 120 + 1 = 121 = 11^2$ $\text {(3)}: \quad$ $\, \displaystyle \tuple {71, 7}: \,$ $\displaystyle 7! + 1$ $=$ $\displaystyle 5040 + 1 = 5041 = 71^2$

## Source of Name

This entry was named for Pierre René Jean Baptiste Henri Brocard.