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:

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

\(\, \ds \tuple {5, 4}: \, \)

\(\ds 4! + 1\)

\(=\)

\(\ds 24 + 1 = 25 = 5^2\)

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

\(\, \ds \tuple {11, 5}: \, \)

\(\ds 5! + 1\)

\(=\)

\(\ds 120 + 1 = 121 = 11^2\)

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

\(\, \ds \tuple {71, 7}: \, \)

\(\ds 7! + 1\)

\(=\)

\(\ds 5040 + 1 = 5041 = 71^2\)

Also see

• Definition:Brown Numbers

Source of Name

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

Sources

• 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