A note on the ternary Diophantine equation x2 − y2m = zn

Attila Bérczes; Maohua Le; István Pink; Gökhan Soydan

doi:10.2478/auom-2021-0020

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A note on the ternary Diophantine equation x2 − y2m = zn

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 29 (2021): Issue 2 (June 2021)

By: Attila Bérczes, Maohua Le, István Pink and Gökhan Soydan

Open Access

|Jul 2021

Abstract

Let ℕ be the set of all positive integers. In this paper, using some known results on various types of Diophantine equations, we solve a couple of special cases of the ternary equation x² − y^2m = zⁿ, x, y, z, m, n ∈ ℕ, gcd(x, y) = 1, m ≥ 2, n ≥ 3.

References

[1] M.A. Bennett, I. Chen, S.R. Dahmen, S. Yazdani, Generalized Fermat equations: a miscellany, Int. J. Number Theory, 11 (2015), 1-28.10.1142/S179304211530001X
Search in Google Scholar Back to article
[2] M.A. Bennett, P. Mihăilescu, S. Siksek, The generalized Fermat equation, Open Problems in Mathematics, (J. F. Nash, Jr. and M. Th. Rassias eds.), Springer, New York (2006), 173-205.10.1007/978-3-319-32162-2_3
Search in Google Scholar Back to article
[3] M. A. Bennett and C. M. Skinner, Ternary Diophantine equations via Galois representations and modular forms, Canad. J. Math., 15 (2004), 23-54.10.4153/CJM-2004-002-2
Search in Google Scholar Back to article
[4] Y. Bugeaud, On the Diophantine equation x² 2^m = yⁿ, Proc. Amer. Math. Soc., 125 (1997), 3203-3208.10.1090/S0002-9939-97-04093-8
Search in Google Scholar Back to article
[5] Y. Bugeaud, On the Diophantine equation x² p^m = yⁿ, Acta Arith., 80 (1997), 213-223.10.4064/aa-80-3-213-223
Search in Google Scholar Back to article
[6] J.H.E. Cohn, The Diophantine equation xⁿ = Dy² + 1, Acta Arith., 106 (2003), 73-83.10.4064/aa106-1-5
Search in Google Scholar Back to article
[7] H. Darmon and L. Merel, Ternary Diophantine equations via Galois representations and modular forms, J. Reine Angew. Math., 490 (1997), 81-100.
Search in Google Scholar Back to article
[8] W. Ivorra, Sur les e´quations x^p + 2^βy^p = z² et x^p + 2^βy^p = 2z², Acta Arith., 108 (2003), 327338.10.4064/aa108-4-3
Search in Google Scholar Back to article
[9]C. Ko, On the Diophantine equation x² = yⁿ + 1, xy ≠ 0, Sci. Sinica 14 (1965), 457-460.
Search in Google Scholar Back to article
[10] M.-H. Le, Some exponential Diophantine equations I: The equation D1x² − D2y² = λk^z, J. Number Theory, 55 (1995), 209-221.10.1006/jnth.1995.1138
Search in Google Scholar Back to article
[11] P. Mihăilescu, Primary cyclotomic units and a proof of Catalan’s conjecture, J. Reine Angew. Math.., 572 (2004), 167-195.
Search in Google Scholar Back to article
[12] L. J. Mordell, Diophantine equations, London, Academic Press, 1969.
Search in Google Scholar Back to article
[13] A. Schinzel and R. Tijdeman, On the equation y^m = P (x), Acta Arith., 31 (1976), 199-204.10.4064/aa-31-2-199-204
Search in Google Scholar Back to article
[14] S. Siksek, On the Diophantine equation x² = yp + 2^kz^p, Journal de Théorie des Nombres de Bordeaux, 15(2003), 839-846.10.5802/jtnb.429
Search in Google Scholar Back to article
[15] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math., 141 (1995), 553-572.10.2307/2118560
Search in Google Scholar Back to article
[16] A. Wiles, Modular elliptic curves and Fermat’s last theorem, Ann. of Math., 141 (1995), 443-551.10.2307/2118559
Search in Google Scholar Back to article
[17] H. Yang and R. Fu, An upper bound for least solution of the exponential Diophantine equation D1x² D₂y² = λk^z, Int. J. Numb. Theo., 11 (2015), 11071114.10.1142/S1793042115500608
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/auom-2021-0020 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

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Language: English

Page range: 93 - 105

Submitted on: Nov 20, 2020

Accepted on: Dec 30, 2020

Published on: Jul 8, 2021

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

higher Diophantine equation,

exponential Diophantine equation,

generalized Fermat equation,

ternary equation

Related subjects:

Mathematics,

General mathematics

© 2021 Attila Bérczes, Maohua Le, István Pink, Gökhan Soydan, published by Ovidius University of Constanta
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 29 (2021): Issue 2 (June 2021)