Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-04T17:51:00.041Z Has data issue: false hasContentIssue false
  • English
  • Français

A GENERALIZATION OF THE RAMANUJAN–NAGELL EQUATION

Part of: Diophantine equations

Published online by Cambridge University Press:  22 August 2018

TOMOHIRO YAMADA Open the ORCID record for TOMOHIRO YAMADA [Opens in a new window]
TOMOHIRO YAMADA*
Affiliation:
Center for Japanese Language and Culture, Osaka University, 8-1-1, Aomatanihigashi, Minoo, Osaka 562-8558, Japan e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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}.

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11D45: Counting solutions of Diophantine equations
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 61 , Issue 3 , September 2019 , pp. 535 - 544
Copyright
Copyright © Glasgow Mathematical Journal Trust 2018 

References

REFERENCES

Apéry, R., Sur une équation diophantinne, C. R. Acad. Sci. Paris, Sér. A 251 (1960), 1451–1452.Google Scholar
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
Bennett, M. A. and Skinner, C. M., Ternary diophantine equations via Galois representations and modular forms, Can. J. Math. 56 (2004), 23–54.CrossRefGoogle Scholar
Bérczes, A. and Pink, I., On generalized Lebesgue-Ramanujan–Nagell equations, An. Şt. Univ. Ovivius Constanţa 22 (2014), 51–71.Google Scholar
Beukers, F., On the generalize Ramanujan–Nagell equation I, Acta Arith. 38 (1980/81), 389–410.CrossRefGoogle Scholar
Beukers, F., On the generalize Ramanujan–Nagell equation II, Acta Arith. 39 (1981), 123–132.CrossRefGoogle Scholar
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
Bugeaud, Y., On some exponential diophantine equations, Monatsh. Math. 132 (2001), 93–97.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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
Cohn, J. H. E., The diophantine equation x 2 + C = yn, Acta Arith. 65 (1993), 367–381.CrossRefGoogle Scholar
Evertse, J.-H., On equations in S-units and the Thue–Mahler equation, Inv. Math. 75 (1984), 561–584.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
Hu, Y. and Le, M., On the number of the generalized Ramanujan–Nagell equation D 1 x + D 2 Sm = pn, Bull. Math. Soc. Sci. Math. Roumanie 55 (2012), 279–293.Google Scholar
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
Le, M., The diophantine equation x 2 + Dm = pn, Acta Arith. 52 (1989), 225–235.CrossRefGoogle Scholar
Le, M., Diophantine equation x 2 + 2m = yn, Chinese Sci. Bull. 42 (1997), 1515–1517.Google Scholar
Le, M., On the diophantine equation x 2 + D = 4pn, J. Number Theory 41 (1997), 87–97.Google Scholar
Le, M., On the diophantine equation D 1x 2 + D 2 = 2n+2, Acta Arith. 64 (1993), 29–41.CrossRefGoogle Scholar
Le, M., A note on the number of solutions of the generalized Ramanujan–Nagell equation D 1x 2 + D 2 = 4pn, J. Number Theory 62 (1997), 100–106.CrossRefGoogle Scholar
Le, M., On the diophantine equation (x 3 - 1)/(x - 1) = (yn - 1)/(y - 1), Trans. Amer. Math. Soc. 351 (1999), 1063–1074.CrossRefGoogle Scholar
Lebesgue, M., Sur l'impossibilité, en nombres entiers, de l'équation xm = y 2 + 1, Nouv. Ann. Math. 9 (1850), 178–180.Google Scholar
Leu, M.-G. and Li, G.-W., The diophantine equation 2x 2 + 1 = 3n, Proc. Amer. Math. Soc. 131 (2003), 3643–3645.CrossRefGoogle Scholar
Ljunggren, W., Noen setninger om ubestemte likninger av formen $\frac{x^n-1}{x-1}=y^q$, Norsk. Mat. Tidsskr. 25 (1943), 17–20.Google Scholar
Nagell, T., Sur l'impossibilité de quelques équations a deux indetérminées, Norsk Mat. Foreninngs Skrifter I Nr. 13 (1923), 1–18.Google Scholar
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
Pink, I., On the diophantine equation x 2 + 2α3β5γ7δ = yn, Publ. Math. Debecen 70 (2007), 149–166.Google Scholar
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
Skinner, C., The diophantine equation x 2 =4qn - 4q + 1, Pacific J. Math. 139 (1989), 303–309.CrossRefGoogle Scholar
Skolem, T., The use of p-adic method in the theory of diophantine equations, Bull. Soc. Math. Belg. 7 (1955), 83–95.Google Scholar
Taylor, R. and Wiles, A., Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (1995), 553–572.CrossRefGoogle Scholar
Wiles, A., Modular elliptic curves and Fermat's last theorem, Ann. Math. 141 (1995), 443–551.CrossRefGoogle Scholar
Yuan, P. and Hu, Y., On the diophantine equation x 2 + Dm = pn, J. Number Theory 111 (2005), 144–153.CrossRefGoogle Scholar