@ChrisWuthrich I agree. I'm interested in this because of a theorem by Silverman which gives an explicit upper bound for the number of integral solutions for these curves.

ams.org/journals/jams/2006-19-03/S0894-0347-06-00515-7/…, I believe section 4 of might have what you need.

OEIS A053669 gives a very unique closed form expression, which is reproduced below. $$a(n) = 1 + \sum_{k=1}^{n} \left( \left\lfloor \frac{n^k}{k!} \right\rfloor - \left\lfloor \frac{n^k - 1}{k!} \right\rfloor \right)$$

I'm trying to solve a question that actually needs the most explicit bound possible, the best I've found is $10h_{3}(-108d)$ where $h_3(-108d)$ is the class number for cubic form with a discriminant $d$ — and since we don't have an estimate on the bound of the rank of elliptic curves, I don't know what to do. In case you're interested, the actual problem I'm solving is how many elliptic curves exist such that $y^2 = x^3 + D$ has exactly $D$ solutions

Thank you so much. I have one last question: Do you know of any papers where an explicit bound for the number of solutions is given in terms of the discriminant (apart from the one I mentioned)? Either way, thank you very much for your response!

@2734364041 Thank you for your response. Two questions: Firstly, isn't theorem 1.2 related to the number of $S$ integral points over an elliptic curve instead of just the integral points? Secondly, why do people believe $\mathrm{rank}(E_{A,B})<c$? Didn't Prof. Elkies discover a curve with rank $28$?

@ChrisWuthrich Thank you for your comment - I'm not looking at a specific example however, i'm looking at the general class of mordell equations

Yes, I need to know the explicit bound $C$ in order to do some computations about specific solutions to Mordell Equations. If I have to find the constant, how would I go about doing it? Also, what evidence points to $C$ being effectively computable? I know it doesn't depend upon Roth's theorem, which is a good sign..

I've edited the original question to reflect the change in the question if that's alright

Thank you so much. How exactly should I decide which $C$ to use? Is this related to Hall's conjecture?