Abstract
By the theory of elliptic curves, we prove that the Diophantine equations \(z^2=f(x)^2 \pm f(y)^2\) have infinitely many rational solutions for some quartic polynomials, which gives a positive answer to Question 4.3 of Ulas and Togbé (Publ Math Debrecen 76(1–2):183–201, 2010) for quartic polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symb. Comput. 24, 235–265 (1997)
H. Cohen, Number Theory, Vol. I: Tools and Diophantine Equations, Graduate Texts in Mathematics, vol. 239 (Springer, New York, 2007)
B. He, A. Togbé, M. Ulas, On the Diophantine equation \(z^2=f(x)^2\pm f(y)^2\), II. Bull. Aust. Math. Soc. 82(2), 187–204 (2010)
L.J. Mordell, Diophantine Equations, Pure and Applied Mathematics, vol. 30 (Academic, London, 1969)
J.H. Silverman, J. Tate, Rational Points on Elliptic Curves (Springer, New York, 1992)
Sz Tengely, M. Ulas, On certain Diophantine equations of the form \(z^2=f(x)^2\pm g(y)^2\). J. Number Theory 174, 239–257 (2017)
M. Ulas, A. Togbé, On the Diophantine equation \(z^2=f(x)^2\pm f(y)^2\). Publ. Math. Debrecen 76(1–2), 183–201 (2010)
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the National Natural Science Foundation of China (Grant No. 11501052).
Rights and permissions
About this article
Cite this article
Zhang, Y., Zargar, A.S. On the Diophantine equations \(z^2=f(x)^2 \pm f(y)^2\) involving quartic polynomials. Period Math Hung 79, 25–31 (2019). https://doi.org/10.1007/s10998-018-0259-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-018-0259-7