10
$\begingroup$

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.

$\endgroup$

1 Answer 1

13
$\begingroup$

This is completely worked out in Walsh's paper:

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.

$\endgroup$
2
  • $\begingroup$ 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 $k\leq 3$? $\endgroup$
    – Benjamin
    Commented Oct 31, 2015 at 14:06
  • $\begingroup$ @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). $\endgroup$
    – Myshkin
    Commented Oct 31, 2015 at 14:12

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.