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SOLUTIONS TO A LEBESGUE–NAGELL EQUATION

Part of: Diophantine equations

Published online by Cambridge University Press:  24 May 2021

NGUYEN XUAN THO Open the ORCID record for NGUYEN XUAN THO [Opens in a new window]
NGUYEN XUAN THO*
Affiliation:
School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet Road, Hanoi, Vietnam
*
e-mail: tho.nguyenxuan1@hust.edu.vn
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Abstract

We find all integer solutions to the equation $x^2+5^a\cdot 13^b\cdot 17^c=y^n$ with $a,\,b,\,c\geq 0$ , $n\geq 3$ , $x,\,y>0$ and $\gcd (x,\,y)=1$ . Our proof uses a deep result about primitive divisors of Lucas sequences in combination with elementary number theory and computer search.

Keywords

Diophantine equationLebesgue–Nagell equationprimitive divisor theorem

MSC classification

Primary: 11D61: Exponential equations
Secondary: 11D72: Equations in many variables
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 105 , Issue 1 , February 2022 , pp. 19 - 30
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Copyright
© 2021 Australian Mathematical Publishing Association Inc.

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Footnotes

The author is partially supported by the Vietnam Institute for Advanced Study in Mathematics (VIASM) and the Vietnam National Foundation for Science and Technology Development (NAFOSTED) (grant number 101.04-2019.314).

References

Alan, M. and Zengin, U., ‘On the Diophantine equation ${x}^2+{3}^a\cdot 4{1}^b={y}^n$ ’, Period. Math. Hungar. 81 (2020), 284–291.CrossRefGoogle Scholar
Arif, S. A. and Muriefah, F. S. A., ‘On the Diophantine equation ${x}^2+{q}^{2k+1}={y}^n$ ’, J. Number Theory 95 (2002), 95–100.CrossRefGoogle Scholar
Bérczes, A. and Pink, I., ‘On the Diophantine equation ${x}^2+{p}^{2k}={y}^n$ ’, Arch. Math. 91 (2008), 505–517.CrossRefGoogle Scholar
Bérczes, A. and Pink, I., ‘On the Diophantine equation ${x}^2+{d}^{2l+1}={y}^n$ ’, Glasg. Math. J. 54 (2012), 415–428.CrossRefGoogle Scholar
Bilu, Y., Hanrot, G. and Voutier, P. M., ‘Existence of primitive divisors of Lucas and Lehmer numbers. With an appendix by M. Mignotte’, J. reine angew. Math. 539 (2001), 75–122.Google Scholar
Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system. I. The user language’, J. Symbolic Comput. 24(3–4) (1997), 235–265.CrossRefGoogle Scholar
Bugeaud, Y., Mignotte, M. and Siksek, S., ‘Classical and modular approaches to exponential Diophantine equations II: The Lebesgue–Nagell equation’, Compos. Math. 142 (2006), 31–62.CrossRefGoogle Scholar
Cangül, I. N., Demirci, M., Inam, I., Luca, F. and Soydan, G., ‘On the Diophantine equation ${x}^2+{2}^a\cdot {3}^b\cdot 1{1}^c={y}^n$ ’, Math. Slovaca 63(3) (2013), 647–659.CrossRefGoogle Scholar
Chakraborty, K., Hoque, A. and Sharma, R., ‘On the solutions of certain Lebesgue– Ramanujan–Nagell equations’, Rocky Mountain J. Math., to appear.Google Scholar
Cohn, J. H. E., ‘The Diophantine equation ${x}^2+C={y}^n$ ’, Acta. Arith. 65 (1993), 367–381.CrossRefGoogle Scholar
Godinho, H., Marques, D. and Togbé, A., ‘On the Diophantine equation ${x}^2+C={y}^n$ for $C={2}^a\cdot {3}^b\cdot 1{7}^c$ and $C={2}^a\cdot 1{3}^b\cdot 1{7}^c$ ’, Math. Slovaca 66 (3) (2016), 565–574.CrossRefGoogle Scholar
Goins, E., Luca, F. and Togbé, A., ‘On the Diophantine equation ${x}^2+{2}^{\alpha}\cdot {5}^{\beta}\cdot 1{3}^{\gamma }={y}^n$ ’, in: Algorithmic Number Theory, 8th International Symposium, ANTS-VIII, Banff, Canada, May 17–22, 2008 (eds. van der Poorten, A. J. and Stein, A.) (Springer, Berlin–Heidelberg, 2008), 430–442.Google Scholar
Le, M. and Soydan, G., ‘A brief survey on the generalized Lebesgue–Ramanujan–Nagell equation’, Surveys in Mathematics and its Applications 15 (2020), 473–523.Google Scholar
Muriefah, F. S. A., Luca, F. and Togbé, A., ‘On the Diophantine equation ${x}^2+{5}^a\cdot 1{3}^b={y}^n$ ’, Glasg. Math. J. 50 (2008), 175–181.Google Scholar
Pink, I., ‘On the Diophantine equation ${x}^2+{2}^a\cdot {3}^b\cdot {5}^c\cdot {7}^d={y}^n$ ’, Publ. Math. Debrecen 70(1–2) (2007), 149–166.CrossRefGoogle Scholar
Pink, I. and Rábai, Z., ‘On the Diophantine equation ${x}^2+{5}^k\cdot 1{7}^l={y}^n$ ’, Commun. Math. 19 (2011), 1–19.Google Scholar
Xiaowei, P., ‘The exponential Lebesgue–Nagell equation ${x}^2+{p}^{2m}={y}^n$ ’, Period. Math. Hungar. 67(2) (2013), 231–242.CrossRefGoogle Scholar
Zhu, H. L. and Le, M. H., ‘On some generalized Lebesgue–Nagell equations’, J. Number Theory 131 (2011), 458–469.CrossRefGoogle Scholar