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)=au3+bu2v+cuv2+dv3 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.
Referring to a theorem of A. E. Ingham, that for all N≥N0 (an absolute constant), the inequality N3≤p≤(N+1)3 is solvable in a prime p, we point out in this paper that it is implicit that he has actually proved that π(x+h)−π(x)∼h(logx)−1 where h=xc and c(>58) is any constant. Further, we point out that even this stronger form can be proved without using the functional equation of ζ(s).