TITLE: Uniform bounds on the number of integer points on a family of elliptic curves
QUESTION [6 upvotes]: Let $P(x,y)$ be a binary cubic polynomial with integer coefficient. Let $n$ be an integer. Suppose the (complex) curve $P(x,y)=n$ is nonsingular, so is an elliptic curve. Is there any bound on the number of the integer points on it that is uniform in $n$? I'm particularly interested in the case when $P(x,y)=xy(x+y)$ (and $n$ is a nonzero integer).

REPLY [6 votes]: As the other answers have made clear, there is a big difference if $n$ is arbitrary, or if $n$ is cube-free. But actually, the right formulation in the latter case is to instead ask for solutions $(x,y)$ satisfying $\gcd(x,y)=1$. So for any $n\in\mathbb Z\setminus\{0\}$, let
$$ N^*(F,n) 
  = \#\bigl\{(x,y)\in\mathbb Z^2 : F(x,y)=n~\text{and}~\gcd(x,y)=1\bigr\}.
$$
Also let $E_{F,n}:F(x,y)=nz^3$ be the associated elliptic curve. Then there is a bound of the form [1]
$$ N^*(F,n) \le C(F)^{1+\text{rank}\,E_{F,n}(\mathbb Q)}, $$
where $C(F)$ is independent of $n$. In particular, if you could find any one $F$, for example $F(x,y)=xy(x+y)$ as you specified, for which you could prove that 
$$ \sup_{n\in\mathbb Z\setminus\{0\}} N^*(F,n) = \infty, $$
then you would also have proven that 
$$ \sup_{n\in\mathbb Z\setminus\{0\}} \text{rank}\,E_{F,n}(\mathbb Q) = \infty, $$
which would be a spectacular result. I've listed a few references that deal with uniform bounds for integer points on (minimal) affine models of elliptic curves. The following MO questions/answers also seem related/relevant:
unboundedness of number of integral points on elliptic curves?
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z[t]$ be unbounded as n varies?
[1] J.H. Silverman, Integer points and the rank of Thue elliptic
curves Invent. Math. 66 (1982), 395-404.
[2] D. Abramovich, Uniformity of stably integral points on elliptic
curves, Invent. Math. 127 (1997), 307-317.
[3] J.H. Silverman, A quantitative version of Siegel's theorem: Integral
points on elliptic curves and Catalan curves, J. Reine Angew. Math.
378 (1987), 60-100.
[4] M. Hindry, J.H. Silverman, The canonical height and integral
points on elliptic curves, Invent. Math. 93 (1988), 419--450.