Results of Brocard-Ramanujan problem on diophantine equation $n!+1=m^2$

TL;DR

This work analyzes the Brocard-Ramanujan problem $n!+1=m^2$, presenting a rigorous framework that ties solvability to the floor of $\sqrt{n!}$ via $m=[\sqrt{n!}]+1$ and the factorization $n!=k(k+2)$. It establishes that any solution must satisfy a precise relation $\varepsilon(2k+\varepsilon)=2k$ with $\sqrt{n!}=k+\varepsilon$, and derives extensive algebraic consequences linking $k$, $\varepsilon$, and auxiliary decompositions so as to bound possible solutions. The paper proves the finiteness of solutions without invoking conjectures, and shows that the known solutions $(n,m)=(4,5),(5,11),(7,71)$ exhaust all feasible cases under its stronger structural conditions. It further constrains the hypothetical fourth solution, arguing that any such $m$ must lie in specific congruence classes and be of enormous magnitude, making a new solution effectively intractable. Overall, the results provide a comprehensive, non-conjectural finiteness result and a detailed map of the arithmetic structure governing potential solutions.

Abstract

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 purpose of this study is to show that the solutions should satisfy some necessary and/or sufficient conditions. If $\sqrt{n!}=k+ε,~n>1,~0<ε<1$; then it has solution if and only if $n!=k(k+2)$ and $ε,~k$ are strictly monotonic increasing. It has only finitely many solutions which is not based on any conjecture or previous research on the Brocard-Ramanujan problem. For the new solution of Brocard-Ramanujan problem ($n\ge 10^5$), the value of $ε$ should be more than $0.999 \cdots 905915$ (digit 0 is coming after 228287 numbers of 9 digit, which takes more than 66 pages in (LibreOffice Writer) indicating almost impossibility of new solution. If we consider $n\geq 10^9$, I am unable to calculate the said numbers of 9 digit in the value of $ε$ in my personal laptop (with 8GB Ram) using MATHEMATICA 8. Finally, it has been claimed to discover that the problem has no further solution.