Some results of Brocard-Ramanujan problem on diophantine equation n!+1=m²

2020/04/09 by Maiti, S.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2004.09256

Abstract

The Brocard-Ramanujan problem pertaining to the diophantine equation n!+1=m², 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 \citeBerndt had not found any new solution in 2000 by extensive computer search for a solution with n up to 10⁹. 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.

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