On Polynomials That Are Sums of Two Cubes

Abstract

It is proved that, if F (x) be a cubic polynomial with integral coefficients having the property that F (n) is equal to a sum of two positive integral cubes for all sufficiently large integers n, then F (x) is identically the sum of two cubes of lin ear polynomials with integer coefficients that are positive for sufficiently large x. A similar result is proved in the case where F (n) is merely assumed to be a sum of two integral cubes of either sign. It is deduced that analogous propositions are true for cubic polynomials F (x0 , . . . , xr ) in more than one indeterminate.