TITLE: Variant of Fermat's last theorem
QUESTION [7 upvotes]: By Fermat's last theorem, the equation $u^3+v^3=w^3$ has no solutions
in positive integers $u,v,w$. Now consider the following variant : call $\rho(x)$
the distance between $x$ and the nearest integer, for any real number $x$
(thus $\rho(3)=0,\rho(3.2)=0.2,\rho(3.5)=0.5, \rho(3.9)=0.1$ etc). 
An "approximate" version of the Fermat equation is to ask
for $\rho({( u^3+v^3 )}^{\frac{1}{3}})$ to be arbitrarily small. A trivial
way to achieve this is to make one of the variables very small
compared to the other, say $v$ very small compared to $u$, so that
${( u^3+v^3 )}^{\frac{1}{3}}$ is very near to $u$.
It is therefore natural to ask if there is an absolute constant $C>0$
such that $\rho({( u^3+v^3 )}^{\frac{1}{3}})$ can be made arbitrarily small
with $u,v$ positive and $u \leq C v, v \leq C u$ (so that neither of
$u$ or $v$ dominates).
Can a (reasonably) explicit $(u_n,v_n)$ sequence be found, such that
$\rho(u_n^3+v_n^3)$ tends to $0$ as $n$ goes to infinity and
$u_n \leq C v_n, v_n \leq C u_n$ ?  "Closed-form" formula
would be the best, of course, but even a simple recursion would be nice.

REPLY [15 votes]: Elkies' paper "Rational points near curves and small nonzero |x^3-y^2| via lattice reduction" is relevant to your question:  he discusses the efficient computation of solutions to |x^3 + y^3 - z^3| < M, and remarks that there are conjectured to be infinitely many of these "near-Fermat" triples. Indeed, his argument is very general and is not restricted to algebraic curves:  he finds, for instance, that
2063^pi + 8093^pi = 8128^pi + .019....!