TITLE: Integral points on elliptic curves of the form $y^2=x^3+px$
QUESTION [10 upvotes]: 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.

REPLY [13 votes]: 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.