Mersenne version of Brocard-Ramanujan equation

Year 2023, Volume: 12 Issue: 1 , 22 - 26 , 30.04.2023

Seyran İbrahimov , Ayşe Nalli

https://doi.org/10.54187/jnrs.1219721

https://izlik.org/JA96GX45WH

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

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