Mersenne version of Brocard-Ramanujan equation

Author:

İBRAHİMOV Seyran1ORCID,NALLİ Ayşe2ORCID

Affiliation:

1. KARABÜK ÜNİVERSİTESİ

2. KARABUK UNIVERSITY

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 $n!+1$.

Publisher

Gaziosmanpasa University

Reference15 articles.

1. L. J. Mordell, Diophantine equations, Academic press, 1969.

2. T. Andreescu, D. Andrica, I. Cucurezeanu, An introduction to Diophantine equations, New York: Birkhäuser, 2010.

3. S. Ramanujan, Question 294, Journal of the Indian Mathematical Society 3 (1911) 128.

4. A. Gérardin, Contribution à létude de léquation $1. 2. 3. 4\ldots z+ 1= y^ 2$, Nouvelles annales de mathématiques 6 (1906) 222-226.

5. B. C. Berndt, W. F. Galway, On the Brocard-Ramanujan Diophantine equation $n! + 1= m^2$, The Ramanujan Journal 4 (1) (2000) 41-42.

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