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Mersenne version of Brocard-Ramanujan equation
2023
Journal:  
Journal of New Results in Science
Author:  
DOI:  
10.54187/jnrs.1219721
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:

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2023
Journal:  
Journal of New Results in Science
Author:  
DOI:  
10.54187/jnrs.1219721
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Journal of New Results in Science

Journal Type :   Uluslararası

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Article : 261
Cite : 245
2023 Impact : 0.087
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Journal of New Results in Science