As the title says. Can we determine all the integral points on elliptic curves of the form
$$y^2=x^3+px$$
for a prime $p$? If yes, can someone explain me how? A good reference would also be sufficient.
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This is completely worked out in Walsh's paper:
- P. G. Walsh, Integer Solutions to the Equation $y^2=x(x^2\pm p^k)$ (2008)
In particular, for your elliptic curve $y^2=x^3+px$, there are at most 2 primitive integer solutions if $p>3$, and 4 if $p=3$.
As for imprimitive solutions in this case ($k=1$, $d=p$), the proof of theorem 4 indicates that such a solution would be a counterexample to the Ankeny-Artin-Chowla conjecture.
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$\begingroup$ Thanks a lot. But it seems that there is no explanation how to determine the imprimitive solutions of $k=1$. In section $7$ on page 1299 he explain how to compute imprimitive solutions only for $k>3$. Does that mean that there are no imprimitive solutions for $k\leq 3$? $\endgroup$– BenjaminCommented Oct 31, 2015 at 14:06
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$\begingroup$ @Marc Look at the last paragraph of page 1297 and the first one of 1298. I think the conclusion is that there is no solution if and only if the Ankeny-Artin-Chowla conjecture is true, so it is an open problem (but most likely true). $\endgroup$– MyshkinCommented Oct 31, 2015 at 14:12