Consider the Mordell equation x^3 – y^2 = k, where x is a non-square positive integer and y^2 is the perfect square nearest to x^3. Noam Elkies (see http://www.math.harvard.edu/~elkies/hall.html) found in 1998 that there are only 25 integers below 10^18 for which |k| < sqrt(x) (the first three being 2, 5234 and 8158). For these 25 numbers (and for the approximately 30 larger numbers found to date) the separation is huge.

I am not aware of any research results that prove a minimum separation between such numbers in general: If x is such a number, what is the minimum separation between x and the next highest such number? Is anyone aware of such research and/or able to comment on how simple/difficult this question is? The question obviously also relates to the separation between lattice points on the Mordell elliptic curve for large x (k fixed).

Elkies (Rational points near curves and small nonzero |x^3 − y^2| via lattice reduction, May 2000) has proved an upper bound (order of sqrt(N).log(N)) for the number of such points not exceeding N, but it is not clear to me whether his method implies any minimum separation. I would greatly appreciate any information on the existence or otherwise of research on this question.

gap principlesort of statement. I'll have to think about it. I wrote a paper with a general gap principle for integral points on elliptic curves, but I'm not sure if it's relevant here. OTOH, if the specific question is about the $x\in\mathbb{N}$ such that there is a $y$ satisfying $0<|x^3-y^2|<\sqrt{x}$, it's not at all clear (at least to me) that the set of such $x$ is infinite. But maybe one could fix a small $\epsilon$ and take $x$ values admitting a solution to $0<|x^3-y^2|<x^{1/2+\epsilon}$, or use an upper bound of $x^{1/2}(\log x)^k$ for some fixed $k$. $\endgroup$ – Joe Silverman Aug 24 '14 at 23:13J. Reine Angew. Math.378(1987), 60-100. For the curves $E_k:Y^2=X^3+k$ with $k$ 6'th power free, I prove that there is anabsoluteconstant $C$ so that $|E_k(\mathbb{Z})|\le C^{1+rank~E_k(\mathbb{Q})}$`. $\endgroup$ – Joe Silverman Aug 25 '14 at 12:46