Geodesic
Some observations on the Diophantine equation $f(x)f(y)=f(z)^2$
Yong Zhang1
1 College 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
Colloquium Mathematicum, Tome 142 (2016) no. 2, pp. 275-283

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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.
DOI : 10.4064/cm142-2-8
Keywords: mathbb polynomial without multiple roots mathop deg geq conditions diophantine equation y has infinitely many nontrivial integer solutions prove equation has rational parametric solution infinitely many irreducible cubic polynomials moreover consider y quartic polynomials

Yong Zhang 1

1 College 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 
@article{10_4064_cm142_2_8,
     author = {Yong Zhang},
     title = {Some observations on the {Diophantine} equation $f(x)f(y)=f(z)^2$},
     journal = {Colloquium Mathematicum},
     pages = {275--283},
     publisher = {mathdoc},
     volume = {142},
     number = {2},
     year = {2016},
     doi = {10.4064/cm142-2-8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4064/cm142-2-8/}
}
TY  - JOUR
AU  - Yong Zhang
TI  - Some observations on the Diophantine equation $f(x)f(y)=f(z)^2$
JO  - Colloquium Mathematicum
PY  - 2016
SP  - 275
EP  - 283
VL  - 142
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.4064/cm142-2-8/
DO  - 10.4064/cm142-2-8
LA  - en
ID  - 10_4064_cm142_2_8
ER  - 
%0 Journal Article
%A Yong Zhang
%T Some observations on the Diophantine equation $f(x)f(y)=f(z)^2$
%J Colloquium Mathematicum
%D 2016
%P 275-283
%V 142
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.4064/cm142-2-8/
%R 10.4064/cm142-2-8
%G en
%F 10_4064_cm142_2_8
Yong Zhang. Some observations on the Diophantine equation $f(x)f(y)=f(z)^2$. Colloquium Mathematicum, Tome 142 (2016) no. 2, pp. 275-283. doi: 10.4064/cm142-2-8

Cité par Sources :