In my ongoing search for Mordell curves of rank 8 and above I have currently identified 144,499 curves of a type where $k$ is squarefree and $k^2 = 1$ mod $24$.
In each case the x coordinates are square modulo $k$ (in the instances where the inverse modulo is possible to obtain).
Its probably safe then, given the size of the sample, to conjecture that this applies to all such $k$ that meet the above criteria.
My questions are then: Is this conjecture well founded and why this effect should happen.
Kevin.