On polynomials that equal binary cubic forms.

C Hooley (1)

1 University of Whales

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 $x$.

Mots clés: binary cubic forms; incongruent integers; perfect square; Chebotarev's theorem

Domaines: Mathématiques [math]

C Hooley. On polynomials that equal binary cubic forms.. Hardy-Ramanujan Journal, 2006, Volume 29 - 2006, pp.1 - 17. ⟨10.46298/hrj.2006.153⟩. ⟨hal-01111461⟩