Variants of the Brocard-Ramanujan equation
Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 2, pp. 353-363.

Dans cet article, nous étudions quelques variations sur l’équation diophantienne de Brocard-Ramanujan.

In this paper, we discuss variations on the Brocard-Ramanujan Diophantine equation.

DOI : 10.5802/jtnb.631
Omar Kihel 1 ; Florian Luca 2

1 Department of Mathematics Brock University 500 Glenridge Avenue St. Catharines, Ontario Canada L2S 3A1 00000
2 Mathematical Institute UNAM, Campus Morelia Apartado, Postal 27-3 (Xangari), C.P. 58089 Morelia, Michoacán, Mexico
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Omar Kihel; Florian Luca. Variants of the Brocard-Ramanujan equation. Journal de théorie des nombres de Bordeaux, Tome 20 (2008) no. 2, pp. 353-363. doi : 10.5802/jtnb.631. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.631/

[1] D. Berend, C. Osgood, On the equation P(x)=n! and a question of Erdős. J. Number Theory 42 (1992), no. 2, 189–193. | MR | Zbl

[2] D. Berend, J.E. Harmse, On polynomial-factorial diophantine equations. Trans. Amer. Math. Soc. 358 (2005), no 4, 1741–1779. | MR | Zbl

[3] B.C. Berndt, W.F. Galway, On the Brocard-Ramanujan Diophantine equation n!+1=m 2 . The Ramanujan Journal 4 (2000), 41–42. | MR | Zbl

[4] H. Brocard, Question 166. Nouv. Corresp. Math. 2 (1876), 287.

[5] H. Brocard, Question 1532. Nouv. Ann. Math. 4 (1885), 291.

[6] A. Dabrowski, On the diophantine equation n!+A=y 2 . Nieuw Arch. Wisk. 14 (1996), 321–324. | MR | Zbl

[7] P. Erdős, R. Obláth, Über diophantische Gleichungen der Form n!=x p ±y p und n!±m!=x p . Acta Szeged 8 (1937), 241–255. | JFM | Zbl

[8] A. Gérardin, Contribution à l’étude de l’équation 1·2·3·4z+1=y 2 . Nouv. Ann. Math. 6 (4) (1906), 222–226. | EuDML | JFM | Numdam

[9] H. Gupta, On a Brocard-Ramanujan problem. Math. Student 3 (1935), 71. | JFM

[10] E. Landau, Handbuch der Lehre von der verteilung der Primzahlen, 3rd Edition. Chelsea Publ. Co., 1974.

[11] F. Luca, The Diophantine equation P(x)=n! and a result of M. Overholt. Glas. Mat. Ser. III 37 (57) no. 2 (2002), 269–273. | MR | Zbl

[12] H.L. Montogomery, R.C. Vaughan, The large sieve. Mathematika 20 (1973), 119–134. | MR | Zbl

[13] M. Overholt, The diophantine equation n!+1=m 2 . Bull. London Math. Soc. 25 (1993), 104. | MR | Zbl

[14] R. M. Pollack, H. N. Shapiro, The next to last case of a factorial diophantine equation. Comm. Pure Appl. Math. 26 (1973), 313–325. | MR | Zbl

[15] S. Ramanujan, Question 469. J. Indian Math. Soc. 5 (1913), 59.

[16] S. Ramanujan, Collected papers. New York, 1962.

[17] S. Wolfram, Math World. http://mathworld.wolfram.com/WilsonTheorem.html

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