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$%C. Hooley, On polynomials that are sums
of two cubes, Rev. Mat. Complut. 20 (2007), no. 1, 207–238. %$

On Polynomials That Are Sums
of Two Cubes

Christopher HOOLEY

Cardiff University

Senghennydd Road

Cardiff, CF24 4AG — United Kingdom

reynishv@Cardiff.ac.uk

Accepted: October 4, 2006

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
linear 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(x_{0},…,x_{r}) in more than one indeterminate.

Key words: polynomials, cubes.

2000 Mathematics Subject Classification: 11D32, 11P99.