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T. Andreescu, D. Andrica, I. Cucurezeanu, An introduction to Diophantine equations, New York: Birkhäuser, 2010.
S. Ramanujan, Question 294, Journal of the Indian Mathematical Society 3 (1911) 128.
A. Gérardin, Contribution à létude de léquation $1. 2. 3. 4\ldots z+ 1= y^ 2$, Nouvelles annales de mathématiques 6 (1906) 222-226.
B. C. Berndt, W. F. Galway, On the Brocard-Ramanujan Diophantine equation $n! + 1= m^2$, The Ramanujan Journal 4 (1) (2000) 41-42.
D. Marques, The Fibonacci version of the Brocard-Ramanujan Diophantine equation, Portugaliae Mathematica 68 (2) (2011) 185-189.
V. Facó, D. Marques, Tribonacci Numbers and the Brocard-Ramanujan Equation, Journal of Integer Sequences 19 (4) (2016) 1-7.
A. Dabrowski, M. Ulas, Variations on the Brocard-Ramanujan equation, Journal of Number Theory 133 (4) (2013) 1168-1185.
B. Demirtürk, R. Keskin, Integer solutions of some Diophantine equations via Fibonacci and Lucas numbers, Journal of Integer Sequences 12 (8) (2009) 1-14.
N. Y. Özgür, On the sequences related to Fibonacci and Lucas numbers, Journal of the Korean Mathematical Society 42 (1) (2005) 135-151.
J. J. Bravo, F. Luca, K. Yazán, On a problem of Pillai with Tribonacci numbers and powers of 2, Bulletin of the Korean Mathematical Society 54 (3) (2017) 1069-1080.
M. Ddamulira, F. Luca, M. Rakotomalala, On a problem of Pillai with Fibonacci numbers and powers of 2, Proceedings-Mathematical Sciences 127 (3) (2017) 411-421.
F. Erduvan, R. Keskin, Z. Siar, Repdigits base b as products of two Lucas numbers, Quaestiones Mathematicae 44 (10) (2021) 1283-1293.
G. H. Hardy, J. E. Littlewood, G. Pólya, G. Pólya, Inequalities, Cambridge, 1952.
R. J. Friedlander, Factoring factorials, The Two-Year College Mathematics Journal 12 (1) (1981) 12-20.
Mersenne version of Brocard-Ramanujan equation
Year 2023,
Volume: 12 Issue: 1, 22 - 26, 30.04.2023
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 $n!+1$.
L. J. Mordell, Diophantine equations, Academic press, 1969.
T. Andreescu, D. Andrica, I. Cucurezeanu, An introduction to Diophantine equations, New York: Birkhäuser, 2010.
S. Ramanujan, Question 294, Journal of the Indian Mathematical Society 3 (1911) 128.
A. Gérardin, Contribution à létude de léquation $1. 2. 3. 4\ldots z+ 1= y^ 2$, Nouvelles annales de mathématiques 6 (1906) 222-226.
B. C. Berndt, W. F. Galway, On the Brocard-Ramanujan Diophantine equation $n! + 1= m^2$, The Ramanujan Journal 4 (1) (2000) 41-42.
D. Marques, The Fibonacci version of the Brocard-Ramanujan Diophantine equation, Portugaliae Mathematica 68 (2) (2011) 185-189.
V. Facó, D. Marques, Tribonacci Numbers and the Brocard-Ramanujan Equation, Journal of Integer Sequences 19 (4) (2016) 1-7.
A. Dabrowski, M. Ulas, Variations on the Brocard-Ramanujan equation, Journal of Number Theory 133 (4) (2013) 1168-1185.
B. Demirtürk, R. Keskin, Integer solutions of some Diophantine equations via Fibonacci and Lucas numbers, Journal of Integer Sequences 12 (8) (2009) 1-14.
N. Y. Özgür, On the sequences related to Fibonacci and Lucas numbers, Journal of the Korean Mathematical Society 42 (1) (2005) 135-151.
J. J. Bravo, F. Luca, K. Yazán, On a problem of Pillai with Tribonacci numbers and powers of 2, Bulletin of the Korean Mathematical Society 54 (3) (2017) 1069-1080.
M. Ddamulira, F. Luca, M. Rakotomalala, On a problem of Pillai with Fibonacci numbers and powers of 2, Proceedings-Mathematical Sciences 127 (3) (2017) 411-421.
F. Erduvan, R. Keskin, Z. Siar, Repdigits base b as products of two Lucas numbers, Quaestiones Mathematicae 44 (10) (2021) 1283-1293.
G. H. Hardy, J. E. Littlewood, G. Pólya, G. Pólya, Inequalities, Cambridge, 1952.
R. J. Friedlander, Factoring factorials, The Two-Year College Mathematics Journal 12 (1) (1981) 12-20.
İbrahimov, S., & Nalli, A. (2023). Mersenne version of Brocard-Ramanujan equation. Journal of New Results in Science, 12(1), 22-26. https://doi.org/10.54187/jnrs.1219721
AMA
İbrahimov S, Nalli A. Mersenne version of Brocard-Ramanujan equation. JNRS. April 2023;12(1):22-26. doi:10.54187/jnrs.1219721
Chicago
İbrahimov, Seyran, and Ayşe Nalli. “Mersenne Version of Brocard-Ramanujan Equation”. Journal of New Results in Science 12, no. 1 (April 2023): 22-26. https://doi.org/10.54187/jnrs.1219721.
EndNote
İbrahimov S, Nalli A (April 1, 2023) Mersenne version of Brocard-Ramanujan equation. Journal of New Results in Science 12 1 22–26.
IEEE
S. İbrahimov and A. Nalli, “Mersenne version of Brocard-Ramanujan equation”, JNRS, vol. 12, no. 1, pp. 22–26, 2023, doi: 10.54187/jnrs.1219721.
ISNAD
İbrahimov, Seyran - Nalli, Ayşe. “Mersenne Version of Brocard-Ramanujan Equation”. Journal of New Results in Science 12/1 (April 2023), 22-26. https://doi.org/10.54187/jnrs.1219721.
JAMA
İbrahimov S, Nalli A. Mersenne version of Brocard-Ramanujan equation. JNRS. 2023;12:22–26.
MLA
İbrahimov, Seyran and Ayşe Nalli. “Mersenne Version of Brocard-Ramanujan Equation”. Journal of New Results in Science, vol. 12, no. 1, 2023, pp. 22-26, doi:10.54187/jnrs.1219721.
Vancouver
İbrahimov S, Nalli A. Mersenne version of Brocard-Ramanujan equation. JNRS. 2023;12(1):22-6.