Proof that every natural number is the sum of 9 cubes of natural numbers

Question

What types of proof are there of this result and where can I read about it?

I think that the Hardy-Littlewood circle method can prove that every number is the sum of something like 100000 cubes, and you can use tables to prove those "small" numbers are all expressible as sums of cubes.. which gives you warings problem but I was more interested in specific proof about the cubes.

Dickson showed that the only integers requiring nine cubes are 23 and 239. Wieferich proved that only 15 integers require eight cubes: 15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428, and 454 from mathworld

Answer 1

There is a paper by L E Dickson, Simpler proofs of Waring's Theorem on cubes, with various generalizations, from the Transactions of the American Mathematical Society for 1928, available here. But you may be disappointed --- "Simpler" doesn't mean "simple".