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I am concerned by the number N of integer points in some class of elliptic curves. It is known to be finite for each elliptic curve C the corresponding bound being a function $N_C$ which gives a huge number. But nevertheless not an absolute constant is known bounding all possible N.

QUESTION: In general, given an arbitrary N, there is any elliptic curve having at least N integers points?

NOTE: In my more difficult problem,besides of certain class of curves, should also be certain condition for such integer points, but here I just ask the question above (difficult too). Any comment or answer (?) will be very appreciated.

Piquito
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  • What curve? For some it is possible to record decisions. There is a question to clarify. – individ Apr 25 '15 at 18:31
  • I answered with some detail to you, Individ, but the note has unfortunately disappeared. I have editing something. Thanks. – Piquito Apr 25 '15 at 19:32
  • See http://math.stackexchange.com/q/32847/14699 – Álvaro Lozano-Robledo Apr 25 '15 at 19:34
  • If you assume your curve to be given by a minimal model or some such (to avoid simply scaling rational points to give, supposing positive rank, as many integral points as desired), the current belief is that the number of "integral" points should be absolutely bounded. – Mike Bennett Apr 25 '15 at 23:35
  • @Mike Bennet It is an almost general feeling but far from being sure. Elliptic curves have a lot of mysteries I was "sure" that the curves $X^3$ + $Y^3$ = A $Z^3$ (in which I search) have rank small (Selmer believed this) but there are of rank 12 and credibly more. The examples I am loking for are these last curves but the question here goes for any class (because easier this way). (Not for infinite rational points but finite integers) Sorry for bad english. – Piquito Apr 26 '15 at 00:01
  • @ Álvaro Lozano-Robledo, Thanks for your link I read it and it is not in my viewpoint unless a comment which don't goes by the contrary of what I do. (sorry for english) – Piquito Apr 26 '15 at 00:15
  • This form does not curve. For some special case, we can write the decision. http://math.stackexchange.com/questions/1114766/diophantine-equation-3-rd-degree And generally for this type the number of primitive solutions of course. – individ Apr 26 '15 at 04:45
  • @individ What do you mean by "primitive solutions", please? Are them generators? – Piquito Apr 26 '15 at 21:11
  • @LuisGomezSanchez This means that decisions are not multiples of each other. – individ Apr 27 '15 at 04:56

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Yes, see Silverman and Tate's Rational Points on Elliptic Curves. In short, you can show that the curve $C: x^3 +y^3 = m$ has infinitely many rational solutions for $m = 9$. Take $N$ such solutions $\left(x_1, y_1), ..., (x_N, y_N\right)$.

Note that if $x_i := \frac{a_i}{b_i}, y_i :=\frac{c_i}{d_i}$ in lowest terms with $b_i, d_i >0$, then substituting into the equation for $C$ and clearing denominators gives

$$ d_i^3a_i^3 +b_i^3c_i^2 + 9b_i^3d_i^3 \implies d_i|b_i \ \& \ b_i|d_i \implies d_i = b_i. $$ Let $P_i := (d_1d_2\cdots d_{i-1}a_id_{i+1}\cdots d_N, d_1d_2\cdots d_{i-1}c_id_{i+1}\cdots d_N)$. Then you can verify that $P_i$ is a solution to the curve $C': x^3 + y^3 = 9(d_1d_2\cdots d_N)^3$ for $1\leq i \leq N$. There are a lot of details missing that can all be found in the book mentioned above.

Aurel
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  • The curve $x^3+y^3=9z^3$ has rank equal to $1$ whose generator is the point $(1,2,1)$ and have, as you say, infinitely many rational points. But in homogeneous coordinates all this rational points can be considered "integers" which is not an answer to my question. In fact what it is asked is about the rational $(x,y,z)$ for which $z=1$ and the only integer point for this curve is the above generator. Just one, no more. – Piquito Aug 03 '17 at 20:19
  • Your question was "Given an arbitrary $N$, is there any elliptic curve having at least $N$ integer points". I did answer this question. The points $P_i$ above are integer solutions to the elliptic curve $C'$ for $1 \leq i \leq N$. Since $P_i \neq P_j$ for $i \neq j$ by construction, we've found an elliptic curve with at least $N$ distinct integer points. – Aurel Aug 05 '17 at 17:14
  • Dear friend, an affirmative answer to my question means that the number of integer points of all the elliptic curves is uniformly bounded (that the number of integer points is never greater than a certain constant $C$). In the case of $x^3+y^3=m$ where $m$ is "appropriate" it is asked the $(x,y)$ both integers (non rational non-integer). Regards. – Piquito Aug 07 '17 at 01:09
  • If you are asking for an absolute bound on the number of integer points on any elliptic curve, then your question is unclear and should be edited. To my knowledge, there is no such known bound. The portion of your post that is labeled "QUESTION," is answered above. – Aurel Aug 07 '17 at 14:05
  • "To my knowledge there is no such known bound", like you but the progress in mathematics it is indetenible and I'm not up to date. Thanks . Regards. – Piquito Aug 08 '17 at 15:07