Mersenne version of Brocard-Ramanujan equation

Authors : Seyran Ibrahimov, Ayşe Nalli

Pages : 22-26

Doi:10.54187/jnrs.1219721

View : 22 | Download : 9

Publication Date : 2023-04-30

Article Type : Research

Abstract :In this study, we deal with a special form of the Brocard-Ramanujan equation, which is one of the interesting and still open problems of Diophantine analysis. We search for the positive integer solutions of the Brocard-Ramanujan equation for the case where the right-hand side is Mersenne numbers. By using the definition of Mersenne numbers, appropriate inequalities for the parameters of the equation, and the prime factorization of $n!$ we show that there is no positive integer solution to this equation. Thus, we obtain this interesting result demonstrating that the square of any Mersenne number can not be expressed as $n!+1$.
Keywords : Brocard-Ramanujan equation, Mersenne numbers, Diophantine equations