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Some observations on the Diophantine equation $f(x)f(y)=f(z)^2$

Tom 142 / 2016

Yong Zhang Colloquium Mathematicum 142 (2016), 275-283 MSC: Primary 11D72, 11D25; Secondary 11D41. DOI: 10.4064/cm142-2-8

Streszczenie

Let $f\in \mathbb {Q}[X]$ be a polynomial without multiple roots and with $\mathop{\rm deg}(f)\geq 2$. We give conditions for $f(X)=AX^2+BX+C$ such that the Diophantine equation $f(x)f(y)=f(z)^2$ has infinitely many nontrivial integer solutions and prove that this equation has a rational parametric solution for infinitely many irreducible cubic polynomials. Moreover, we consider $f(x)f(y)=f(z)^2$ for quartic polynomials.

Autorzy

  • Yong ZhangCollege of Mathematics and Computing Science
    Changsha University of Science and Technology
    410114 Changsha, People's Republic of China
    and
    Department of Mathematics
    Zhejiang University
    310027 Hangzhou, People's Republic of China
    e-mail

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