Let f ∈ ℚ [X] be a polynomial without multiple roots and with deg(f) ≥ 2. We give conditions for f(X) = AX² + 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-8,
author = {Yong Zhang},
title = {Some observations on the Diophantine equation f(x)f(y) = f(z)$^2$},
journal = {Colloquium Mathematicae},
volume = {144},
year = {2016},
pages = {275-283},
zbl = {06498819},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-8}
}
Yong Zhang. Some observations on the Diophantine equation f(x)f(y) = f(z)². Colloquium Mathematicae, Tome 144 (2016) pp. 275-283. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm142-2-8/