{"docId":153,"paperId":153,"url":"https:\/\/hrj.episciences.org\/153","doi":"10.46298\/hrj.2006.153","journalName":"Hardy-Ramanujan Journal","issn":"","eissn":"2804-7370","volume":[{"vid":40,"name":"Volume 29 - 2006"}],"section":[],"repositoryName":"Hal","repositoryIdentifier":"hal-01111461","repositoryVersion":1,"repositoryLink":"https:\/\/hal.archives-ouvertes.fr\/hal-01111461v1","dateSubmitted":"2015-03-03 16:13:57","dateAccepted":"2015-06-12 16:05:50","datePublished":"2006-01-01 08:00:00","titles":{"en":"On polynomials that equal binary cubic forms."},"authors":["Hooley, C"],"abstracts":{"en":"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 $x$."},"keywords":[{"en":" Chebotarev's theorem"},{"en":" incongruent integers"},{"en":" perfect square"},{"en":"binary cubic forms"},"[MATH] Mathematics [math]"]}