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Numberphile

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42 is the only remaining (eligible) number below 100 which has not been represented as the sum of three cubes... 33 was cracked by Andrew Booker from the University of Bristol.
More links & stuff in full description below ↓↓↓
The original 33 video: https://youtu.be/wymmCdLdPvM
The 74 cracked video: https://youtu.be/_-M_3oV75Lw
Casting out nines: https://youtu.be/FlndIiQa20o
Andrew Booker's webpage: http://www.bristol.ac.uk/maths/people/andrew-r-booker/overview.html
Some of the papers we referenced...
Undecidability in number theory: http://www-math.mit.edu/~poonen/papers/h10_notices.pdf
The density of zeros of forms for which weak approximation fails: http://eprints.maths.ox.ac.uk/156/
Solutions of the Diophantine Equation: x^3+y^3+z^3=k: https://doi.org/10.1112/jlms/s1-30.1.101
Solutions of the Diophantine Equation x^3 + y^3 = z^3 - d: https://www.jstor.org/stable/2003763
Newer sums of three cubes: https://arxiv.org/abs/1604.07746
Early write-up of Booker's work: https://people.maths.bris.ac.uk/~maarb/papers/cubesv1.pdf
Discuss this video on reddit: https://redd.it/b0ax2o
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