Geodesic
A GENERALIZATION OF THE RAMANUJAN–NAGELL EQUATION
Glasgow mathematical journal, Tome 61 (2019) no. 3, pp. 535-544

Voir la notice de l'article provenant de la source Cambridge University Press

We shall show that, for any positive integer D > 0 and any primes p1, p2, the diophantine equation x2 + D = 2sp1kp2l has at most 63 integer solutions (x, k, l, s) with x, k, l ≥ 0 and s ∈ {0, 2}. 
YAMADA, TOMOHIRO. A GENERALIZATION OF THE RAMANUJAN–NAGELL EQUATION. Glasgow mathematical journal, Tome 61 (2019) no. 3, pp. 535-544. doi: 10.1017/S0017089518000344
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[1] Apéry, R., Sur une équation diophantinne, C. R. Acad. Sci. Paris, Sér. A 251 (1960), 1451–1452. Google Scholar

[2] Bender, E. A. and Herzberg, N. P., Some diophantine equations related to the quadratic form ax 2 + by 2, Studies in Algebra and Number Theory (Academic Press, New York, 1979), 219–272. Google Scholar

[3] Bennett, M. A. and Skinner, C. M., Ternary diophantine equations via Galois representations and modular forms, Can. J. Math. 56 (2004), 23–54. Google Scholar | DOI

[4] Bérczes, A. and Pink, I., On generalized Lebesgue-Ramanujan–Nagell equations, An. Şt. Univ. Ovivius Constanţa 22 (2014), 51–71. Google Scholar

[5] Beukers, F., On the generalize Ramanujan–Nagell equation I, Acta Arith. 38 (1980/81), 389–410. Google Scholar | DOI

[6] Beukers, F., On the generalize Ramanujan–Nagell equation II, Acta Arith. 39 (1981), 123–132. Google Scholar | DOI

[7] Bilu, Y., Hanrot, G. and Voutier, P. M., Existence of primitive divisors of Lucas and Lehmer numbers, J. Reine Angew. Math. 539 (2001), 75–122. Google Scholar

[8] Bugeaud, Y., On some exponential diophantine equations, Monatsh. Math. 132 (2001), 93–97. Google Scholar | DOI

[9] Bugeaud, Y., Mignotte, M. and Siksek, S., Classical and modular approaches eto exponential diophantine equations II. The Lebesgue–Nagell equation, Compos. Math. 142 (2006), 31–62. Google Scholar | DOI

[10] Bugeaud, Y. and Shorey, T. N., On the number of solutions of the generalized Ramanujan–Nagell equation, J. Reine Angew. Math. 539 (2001), 55–74. Google Scholar

[11] Cohn, J. H. E., The diophantine equation x 2 + C = yn, Acta Arith. 65 (1993), 367–381. Google Scholar | DOI

[12] Evertse, J.-H., On equations in S-units and the Thue–Mahler equation, Inv. Math. 75 (1984), 561–584. Google Scholar | DOI

[13] Godinho, H., Marques, D. and Togbé, A., On the diophantine equation x 2 + C = yn for C =2a 3b 17c and C =2a 13b 17c, Math. Slovaca 66 (2016), 565–574. Google Scholar | DOI

[14] Hu, Y. and Le, M., On the number of the generalized Ramanujan–Nagell equation D x + D 2 Sm = pn, Bull. Math. Soc. Sci. Math. Roumanie 55 (2012), 279–293. Google Scholar

[15] Le, M., The divisibility of the class number for a class of imaginary quadratic fields, Kexue Tongbao 32 (1987), 724–727 (in Chinese). Google Scholar

[16] Le, M., The diophantine equation x 2 + Dm = pn, Acta Arith. 52 (1989), 225–235. Google Scholar | DOI

[17] Le, M., Diophantine equation x 2 + 2m = yn, Chinese Sci. Bull. 42 (1997), 1515–1517. Google Scholar

[18] Le, M., On the diophantine equation x 2 + D = 4pn, J. Number Theory 41 (1997), 87–97. Google Scholar

[19] Le, M., On the diophantine equation D x 2 + D = 2n+2, Acta Arith. 64 (1993), 29–41. Google Scholar | DOI

[20] Le, M., A note on the number of solutions of the generalized Ramanujan–Nagell equation D x 2 + D = 4pn, J. Number Theory 62 (1997), 100–106. Google Scholar | DOI

[21] Le, M., On the diophantine equation (x 3 - 1)/(x - 1) = (yn - 1)/(y - 1), Trans. Amer. Math. Soc. 351 (1999), 1063–1074. Google Scholar | DOI

[22] Lebesgue, M., Sur l'impossibilité, en nombres entiers, de l'équation xm = y 2 + 1, Nouv. Ann. Math. 9 (1850), 178–180. Google Scholar

[23] Leu, M.-G. and Li, G.-W., The diophantine equation 2x 2 + 1 = 3n, Proc. Amer. Math. Soc. 131 (2003), 3643–3645. Google Scholar | DOI

[24] Ljunggren, W., Noen setninger om ubestemte likninger av formen , Norsk. Mat. Tidsskr. 25 (1943), 17–20. Google Scholar

[25] Nagell, T., Sur l'impossibilité de quelques équations a deux indetérminées, Norsk Mat. Foreninngs Skrifter I Nr. 13 (1923), 1–18. Google Scholar

[26] Nagell, T., The diophantine equation x 2 + 7 = 2n, Nork. Mat. Tidsskr. 30 (1948), 62–64 (in Norwegian), English version: Ark. Mat. 4 (1960), 185–187. Google Scholar

[27] Pink, I., On the diophantine equation x 2 + 2α3β5γ7δ = yn, Publ. Math. Debecen 70 (2007), 149–166. Google Scholar

[28] Shorey, T. N., Van Der Poorten, A. J., Tijdeman, R. and Schinzel, A., Applications of the Gel'fond–Baker method to diophantine equations, in Transcendence theory: Advances and applications (Academic Press, London, 1977), 59–77. Google Scholar

[29] Skinner, C., The diophantine equation x 2 =4qn - 4q + 1, Pacific J. Math. 139 (1989), 303–309. Google Scholar | DOI

[30] Skolem, T., The use of p-adic method in the theory of diophantine equations, Bull. Soc. Math. Belg. 7 (1955), 83–95. Google Scholar

[31] Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553–572. Google Scholar | DOI

[32] Wiles, A., Modular elliptic curves and Fermat's last theorem, Ann. Math. 141 (1995), 443–551. Google Scholar | DOI

[33] Yuan, P. and Hu, Y., On the diophantine equation x 2 + Dm = pn, J. Number Theory 111 (2005), 144–153. Google Scholar | DOI

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