  The Brocard-Ramanujan problem pertaining to the diophantine equation
$n!+1=m^2$, a famously unsolved problem, deals with finding the integer
solutions to the equation. Nobody has discovered any new solution of the
problem beyond $n=4,~5$ and $7$ although many of us have tried it. Bruce Berndt
and William Galway \cite{Berndt} had not found any new solution in 2000 by
extensive computer search for a solution with $n$ up to $10^9$. The main
purpose of this study is to show that the solutions should satisfy some
necessary and/or sufficient conditions and it has only finitely many solutions
which is not based on any conjecture or previous research on the
Brocard-Ramanujan problem.
