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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.

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    $\begingroup$ To clarify: your question is not about the difference between $x^3$ and $y^2$ (Hall's conjecture), but rather about the difference between those values of $x$ for which the spacing between $x^3$ and its nearest square is small (below $\sqrt{x}$). Is that correct? $\endgroup$ – Lucia Aug 23 '14 at 15:39
  • $\begingroup$ @Lucia That would be a gap principle sort 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 23:13
  • $\begingroup$ Joe & Lucia, thanks for the responses. The question I posed is distinct from (but has obvious implications for) lattice point separation on a Mordell elliptic curve C: x^3 - y^2 = k (when the x-coordinates of the lattice points are > k^2). I was aware of the Hall/Davenport connection but, as Lucia pointed out, my question has to do with gaps between successive non-square x which have the property that the distance between x^3 and the square nearest to x^3 is less than sqrt(x). So, there are two distinct but related questions here, with the first having some implications for the second ...... $\endgroup$ – Ryan D'Mello Aug 25 '14 at 4:48
  • $\begingroup$ @Joe Silverman It is known that there are infinitely many x such that there is a y satisfying 0 < |x^3 - y^2| < sqrt(X). In fact, there is what is known as the Danilov-Elkies infinite family, every element of which has this property. This is discussed in Elkies' paper referenced in my original post. (Also see oeis.org/A200216 for some examples of elements in this family.) I am very interested in learning more about your general gap principle for integral points on elliptic curves. What does it state? $\endgroup$ – Ryan D'Mello Aug 25 '14 at 5:04
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    $\begingroup$ I'd forgotten about that Danilov article. Anyway, I thought about it for a little bit and don't see an immediate way to get a gap estimate using (local) canonical height estimates. Seems like an interesting problem. My article with an elliptic curve gap principle is: A quantitative version of Siegel's theorem: Integral points on elliptic curves and Catalan curves, J. 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 an absolute constant $C$ so that $|E_k(\mathbb{Z})|\le C^{1+rank~E_k(\mathbb{Q})}$`. $\endgroup$ – Joe Silverman Aug 25 '14 at 12:46
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Hall's conjecture says that for every $\epsilon>0$ there is a $C_\epsilon$ such that if $x$ and $y$ are integers with $x^3-y^2\ne0$, then $$ |x^3 - y^2| \ge C_\epsilon \max\{|x^3|,|y^2|\}^{1/6-\epsilon}, $$ so this would imply that the separation indeed gets quite large as $x$ and $y$ increase. The polynomial version, i.e., when $x$ and $y$ are in $\mathbb{C}[T]$, was proven by Davenport with $\epsilon=0$, i.e., if $x^3 - y^2 \ne 0$ then $$ \deg(x^3-y^2) \ge \frac16 \max\{\deg(x^3),\deg(y^2)\} + C $$ for an absolute constant $C$ (in fact, we can use $C=1$). More generally in the polynomial (or function field) case, one can easily prove lower bounds for $\deg(x^n-y^m)$, with analogous conjectures over $\mathbb{Z}$ (or over number fields).


Addendum: As Ryan D'Mello points out, the above doesn't really answer the question, which asks about gaps between $x$ values of solutions to $0<|x^3-y^2|<\sqrt{x}$. However, I think that in order to get a reasonable answer, one will need to assume something like Hall's conjecture. Alternatively, for a given (small) $\epsilon>0$, one might hope to unconditionally prove a gap estimate for the set $$ \bigl\{ x\in\mathbb{Z} : \text{there exists $y\in\mathbb{Z}$ with $|x|^{1/2-\epsilon}<|x^3-y^2|<|x|^{1/2+\epsilon}$} \bigr\}, $$ where the size of the gaps depends on $\epsilon$.

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  • $\begingroup$ @KConrad Thanks for filling in $C$, I'd forgotten the exact value and didn't feel like rederiving, it. Actually, if you take $x$ and $y$ to be $S$-integers in a function field of genus $g$, the constant is something like $-(2g-2+\#S)$. As for the $x^3-y^2\ne0$ condition, I didn't think that it was necessary to repeat it, but I guess it doesn't hurt to put it in. $\endgroup$ – Joe Silverman Aug 23 '14 at 16:25
  • $\begingroup$ If you alow the exponent $1/2+\varepsilon$, then the construction from web.math.pmf.unizg.hr/~duje/pdf/hall3.pdf might be relevant. $\endgroup$ – duje Aug 25 '14 at 17:28

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