On polynomials that are sums of two cubes (Q2472514)

Let \(F(x)\in\mathbb{Z}[x]\) be a cubic polynomial with the property that for sufficiently large \(n\), the integer \(f(n)\) is a sum of two positive cubes. It is proved that one may write \(f(x)= L_1(x)^3+ L_2(x)^3\) with linear polynomials \(L_1(x),L_2(x)\in\mathbb{Z}[x]\), such that \(L_1(n)\) and \(L_2(n)\) are positive for sufficiently large \(n\). The corresponding result for sums of two squares was proved by \textit{H. Davenport}, \textit{D. J. Lewis} and \textit{A. Schinzel} [Acta Arith. 11, 353--358 (1966; Zbl 0139.27101)]. The key first step is to show that \(f\) must be reducible. This is achieved by showing that when \(f\) is irreducible the Diophantine equation \(f(n)= r^3+ s^3\) has \(o(X)\) positive integer solutions with \(n\leq X\).