Can we find such triples systematically , or is it a "lucky strike"?

Question

Here :

https://en.wikipedia.org/wiki/Hall%27s_conjecture

the so-called Hall-conjecture is formulated and an example is given that would require a very small constant in the Hall-conjecture. It contains numbers $x$ and $y$, such that the maginude of $|y^2-x^3|$ is "very small" compared to the magnitude of $x$ and $y$.

Can we construct such "spectacular" triples or find them systematically in a reasonable time ? Or is the triple found just a "lucky strike" ?

Answer 1

It's certainly more than just a "lucky strike".

Noam Elkies's page referenced in that Wikipedia page gives some information:

I have found a new algorithm that finds all solutions of $|k| << x^{1/2}$ (or indeed of $|k| << x$) with $x < N$ in time $O(N^{1/2+o(1)})$. I implemented the algorithm in 64-bit C for $N=10^{18}$, and ran it for almost a month...

See also OEIS sequence A078933 and links there.