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Brocard's problem

Problem Statement

Given $n!+1=m^2$ which pairs $(n,m)$ are the solutions to the given equation.

Known Solutions

\[(n,m)=(4,5),(5,11),(7,71)\]

Are the known solutions, and it was a conjecture of Paul Erdös, that these are the only solutions.

Heuristic arguments

- If $n>3$ then $n^2<n!$ therefore $m>n$

- If $n>2p$ LHS is 1 mod $p^2$ so $m$ is likely $\pm 1$ mod $p^2$ making $m$ spread out as $n$ increases.

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