The problem
Question: find all integer solutions (n, m) of n! + 1 = m². The three known: n = 4 (25 = 5²), n = 5 (121 = 11²), n = 7 (5041 = 71²).
Brocard's Problem
openHistory & significance
Posed by Brocard in 1876, rediscovered by Ramanujan who listed it among his problems. Wilson's theorem guarantees m exists whenever n+1 is prime (Wilson primes aside), but squares demand far more. Searches (Berndt et al., extensive computational sweeps) find nothing beyond n = 7. It sits beside the abc-conjecture in difficulty class: any serious progress would need the deepest Diophantine machinery available.
Still open.
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