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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.

While there were some key number theoretic observations, a computer search was still required. It took 3 weeks (or 15 core-years on a supercomputer) to find this solution. In case you were interested, here is the solution:

(8866128975287528)^3 + (−8778405442862239)^3 + (−2736111468807040)^3 = 33

Why this is significant, from Alex Kontorovich (https://twitter.com/AlexKontorovich/status/1104354243643883520?ref_src=twsrc%5Etfw):

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"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)"

Beyond analytic number theory, finding solutions to Diophantine equations (multivariate polynomials with integer coefficients) has been of interest since the early 1900s. The computational problem of finding such solutions is referred to as Hilbert's 10th Problem. This problem is undecidable; that is, no algorithm exists to solve this problem in general.

https://aperiodical....of-three-cubes/