arXiv:1604.07746math.NT
The search of solutions of the Diophantine equation $x^3 + y^3 + z^3 = k$ for $k<1000$ has been extended with bounds of $|x|$, $|y|$ and $|z|$ up to $10^{15}$. The first solution for $k=74$ is reported. This only leaves $k=33$ and $k=42$ for $k<100$ for which no solution has yet been found. A total of 966 new solutions were found.
Linked by 2 identifiable commenters in 2 comments across 2 threads, from Aug 2017 to Mar 2019.
I was replying that the constants in question include 33, 42 and 74... and realized that I'm late :-) In a more recent survey, though, it is now known that: > 74 = −284650292555885^3 + 66229832190556^3 + 283450105697727^3 [1] [2]. [1] https://arxiv.org/pdf/1604.07746v1.pdf [2]…
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Some additional links: Sander Huisman's 2016 paper about solving the three cubes problem for 74, using a supercomputer in Lyon [0]. The approach for 33 is quite similar, just augmenting the range searched (and some additional tricks to further reduce the search space). Andrew…
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