Let f ∈ ℚ [X] and deg f ≤ 3. We prove that if deg f = 2, then the diophantine equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in ℚ (t). In the case when deg f = 3 and f(X) = X(X²+aX+b) we show that for all but finitely many a,b ∈ ℤ satisfying ab ≠ 0 and additionally, if p|a, then p²∤b, the equation f(x)f(y) = f(z)² has infinitely many nontrivial solutions in rationals.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-1,
author = {Maciej Ulas},
title = {On the diophantine equation f(x)f(y) = f(z)$^2$},
journal = {Colloquium Mathematicae},
volume = {107},
year = {2007},
pages = {1-6},
zbl = {1153.11016},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-1}
}
Maciej Ulas. On the diophantine equation f(x)f(y) = f(z)². Colloquium Mathematicae, Tome 107 (2007) pp. 1-6. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm107-1-1/