42 is the new 33

Additional information

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.

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