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:

$\text {(1)}: \quad$

$\, \ds \tuple {5, 4}: \, $

$\ds 4! + 1$

$=$

$\ds 24 + 1 = 25 = 5^2$

$\text {(2)}: \quad$

$\, \ds \tuple {11, 5}: \, $

$\ds 5! + 1$

$=$

$\ds 120 + 1 = 121 = 11^2$

$\text {(3)}: \quad$

$\, \ds \tuple {71, 7}: \, $

$\ds 7! + 1$

$=$

$\ds 5040 + 1 = 5041 = 71^2$

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

1980: David M. Burton: Elementary Number Theory (revised ed.) ... (previous) ... (next): Chapter $1$: Some Preliminary Considerations: $1.1$ Mathematical Induction: Problems $1.1$: $5 \ \text {(a)}$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $4$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $7$
1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $121$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $4$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $7$
1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $121$

Weisstein, Eric W. "Brocard's Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BrocardsProblem.html