Abstract
Let \(f\in {\mathbb {Q}}[X]\) be a polynomial without multiple roots and \(deg(f)\ge 2\). We consider the Diophantine equation \(f(x)f(y)=f(z^2)\). For two classes of irreducible quadratic polynomials, this equation has infinitely many nontrivial integer solutions, if the corresponding Pell’s equations satisfy a condition. For a special cubic polynomial, it has a one-parameter family of rational solutions. For \(f(X)=X(X^2+X+k)\) and \(X(X^2+kX+1)\) there are infinitely many \(k \in {\mathbb {Q}}\) such that the title equation has rational solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M.A. Bennett, The Diophantine equation \((x^k-1)(y^k-1)=(z^k-1)^t\). Indag. Math. 18, 507–525 (2007)
R.K. Guy, Unsolved Problems in Number Theory, 3rd edn. (Springer Science, Heidelberg, 2004)
S. Katayama, On the Diophantine equation \((x^2+1)(y^2+1)=(z^2+1)^2\). J Math Univ Tokushima 40, 9–14 (2006)
A. Schinzel, W. Sierpinski, Sur l’equation Diophantienne \((x^2-1)(y^2-1)=[((y-x)/2)^2-1]^2\). Elem. Math. 18, 132–133 (1963)
K. Szymiczek, On a Diophantine equation. Elem. Math. 22, 37–38 (1967)
M. Ulas, On the Diophantine equation \(f(x)f(y)=f(z)^2\). Colloq. Math. 107, 1–6 (2007)
M. Ulas, On the Diophantine equation \((x^2+k)(y^2+k)=(z^2+k)^2\). Rocky Mt. J. Math. 38, 2091–2097 (2008)
Y. Zhang, T. Cai, On the Diophantine equation \(f(x)f(y)=f(z^2)\). Publ. Math. 82, 31–41 (2013)
L.J. Mordell, Diophantine Equation (Academic Press, Waltham, 1969)
J.H. Silverman, J. Tate, Rational Points on Elliptic Curves (Springer, New York, 1992)
Acknowledgments
The authors are grateful to the referee for his valuable comments and suggestions, which improved the quality of this note. This research was supported by China National Science Foundation Grant (No. 11351002), Natural Science Foundation of Zhejiang Province (No. LQ13A010012), China Postdoctoral Science Foundation (No. 2012M521155) and Zhejiang Projects for Postdoctoral Research Preferred Funds (No. Bsh1201021)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, Y., Cai, T. A note on the Diophantine equation \(f(x)f(y)=f(z^2)\) . Period Math Hung 70, 209–215 (2015). https://doi.org/10.1007/s10998-014-0068-6
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10998-014-0068-6