#### Mordell’s cube sum problem

In 1957, British-American mathematician Louis Mordell asked whether, given some integer $k$, there are integers $x, y, z$ such that $x^3 + y^3 + z^3 = k$. Like Fermat’s last theorem, this problem is very easily stated but very difficult to explore, much less solve definitively.

Some solutions are easy. When $k = 3$, for instance, there are two simple solutions: $1^3 + 1^3 + 1^3 = 3$ and $4^3 + 4^3 + (-5)^3 = 3$. It is also known that there are no solutions in other cases, including $4, 5, 13, 14$ and others.

#### Huisman’s 2016 results

Significant progress was made in 2016 by Sander Huisman. Employing an algorithm due to Noam Elkies, he found a total of 966 new solutions for $k$ in the range $1 \le k \lt 1000$. To begin with, Huisman found the first known solution for $k = 74$, namely $$74 = (−284650292555885)^3 + 66229832190556^3 + 283450105697727^3.$$ He also found a second known solution for each of these three cases: $$606 = (−170404832787569)^3 + (−16010802062873)^3 + 170451934224718^3,$$ $$830 = (−947922123009026)^3 + (−335912682279105)^3 + 961779444965911^3,$$ $$966 = (−1134209166959435)^3 + 291690681248788^3 + 1127741630138089^3.$$ His work left 13 cases less than 1000 for which no solutions had been found: $k = 33, 42, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975$.

#### Booker’s new result for $k = 33$

The latest news here is that Andrew Booker, who confessed he was inspired by watching the Numberphile video The untracked problem with 33, has found a new solution for $k = 33$, one of the unsolved cases remaining from Huisman’s work. Whereas Huisman employed Elkies’ algorithm, Booker employed a scheme based on the fact that in any solution $k – z^3 = x^3 + y^3$ must have $x + y$ as a factor. Booker found that the running time for this scheme is very nearly linear in the height bound of $x, y, z$.

This is Booker’s result: $$33=8866128975287528^3 +(−8778405442862239)^3 +(−2736111468807040)^3.$$ The run required approximately 15 core-years of computation over three weeks. This leaves $k = 42$ as the only case less than 100 for which no solution is known, and 12 cases less than 1000.

But who knows? Maybe others will soon find solutions for these as well…

For some additional details, see this New Scientist article by Donna Lu.

Added 7 Sep 2019: The case $k = 42$ has now been solved, with solution $42 = (-80538738812075974)^3 + 80435758145817515^3 + 12602123297335631^3$. See the update to this Quanta Magazine article.