## C Hooley - On polynomials that equal binary cubic forms.

hrj:153 - Hardy-Ramanujan Journal, January 1, 2006, Volume 29 - https://doi.org/10.46298/hrj.2006.153
On polynomials that equal binary cubic forms.

Authors: C Hooley

Let $F(x)$ be a cubic polynomial with rational integral coefficients with the property that, for all sufficiently large integers $n,\,F(n)$ is equal to a value assumed, through integers $u, v$, by a given irreducible binary cubic form $f(u,v)=au^3+bu^2v+cuv^2+dv^3$ with rational integral coefficients. We prove that then $F(x)=f(u(x),v(x))$, where $u=u(x), v=v(x)$ are linear binomials in $x$.

Volume: Volume 29
Published on: January 1, 2006
Submitted on: March 3, 2015
Keywords: Chebotarev's theorem, incongruent integers, perfect square,binary cubic forms,[MATH] Mathematics [math]