It was an open question whether 33 could be written as the sum of three cubes. Thanks to Andrew R. Booker, it now isn’t.

\begin{array}{c} (8866128975287528)^3 \\ + \\(-8778405442862239)^3 \\ + \\(-2736111468807040)^3 \\ = \\ 33\end{array}

The “sum of three cubes” problem asks whether, for a given $k$, there are integer solutions to

\[ x^3+y^3+z^3=k \]

The question is which integers are expressible in this form, not whether all integers can. For example, we know integers of the form $k = \pm 4 \pmod{9}$ cannot. The problems dates back to at least 1825.

Andrew has released a paper titled *“Cracking the problem with 33”*, explaining how he found his solution. Even after a decent amount of mathematical insight, the search still took a while: “The total computation used approximately 15 core-years over three weeks of real time.”

Alex Kontorovich explained on Twitter the significance of this progress.

Wow this is big news! The sum of three cubes is the bane of modern analytic number theory; its so embarrassing that we can’t tell basic things like which numbers are represented. For a long time, 33 was the smallest unknown culprit. Now that honor belongs to 42 (last below 100)

The general problem of whether a given number can be written as the sum of three cubes has been proven to be undecidable. Bjorn Poonen’s paper *“Undecidability in number theory”* opens with this fact, and describes quite a few other similarly undecidable questions about numbers.

Booker’s paper might be unique in beginning with the words “Inspired by the Numberphile video, …” The video in question is this one, featuring Tim Browning.

### More information

“Cracking the problem with 33”, by Andrew R. Booker.

Tim Browning’s webpage giving the result.

Post on reddit:r/math 33=8866128975287528^3+(-8778405442862239)^3+(-2736111468807040)^3.

Technical background on the problem from this 2007 article in AMS’s *Mathematics of Computation*: New integer representations as the sum of three cubes.

Poonen’s paper says that it’s undecidable as to whether a general Diophantine equation has solutions in the integers. But it doesn’t look like he claims that the specific problem of writing integers as sums of three cubes is undecidable.