On the diophantine equation $f(x)f(y)=f(z)^2$
Colloquium Mathematicum, Tome 107 (2007) no. 1, pp. 1-6
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
Let $f\in\mathbb Q[X]$ and $\mathop{\rm deg}f\leq 3$.
We prove that if
$\mathop{\rm deg}f=2$, then the diophantine equation
$f(x)f(y)=f(z)^2$ has infinitely many nontrivial solutions in
$\mathbb Q(t)$. In the case when $\mathop{\rm deg}f=3$ and
$f(X)=X(X^2+aX+b)$
we show that for all but finitely many $a,b\in\mathbb Z$ satisfying
$ab\neq 0$ and additionally,
if $p\mid a$, then $p^2\nmid b$,
the equation
$f(x)f(y)=f(z)^2$ has infinitely many nontrivial solutions in
rationals.
Mots-clés :
mathbb mathop deg leq prove mathop deg diophantine equation y has infinitely many nontrivial solutions mathbb mathop deg finitely many mathbb satisfying neq additionally mid nmid equation y has infinitely many nontrivial solutions rationals
Affiliations des auteurs :
Maciej Ulas 1
@article{10_4064_cm107_1_1,
author = {Maciej Ulas},
title = {On the diophantine equation $f(x)f(y)=f(z)^2$},
journal = {Colloquium Mathematicum},
pages = {1--6},
year = {2007},
volume = {107},
number = {1},
doi = {10.4064/cm107-1-1},
language = {fr},
url = {http://geodesic.mathdoc.fr/articles/10.4064/cm107-1-1/}
}
Maciej Ulas. On the diophantine equation $f(x)f(y)=f(z)^2$. Colloquium Mathematicum, Tome 107 (2007) no. 1, pp. 1-6. doi: 10.4064/cm107-1-1
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