This is more a question on History than proof itself. About a decade ago, a college professor and a Math coach told us about this beautiful theorem:
Every multiple of 6 can be written as a sum of four cubes
The proof of the theorem is elementary as well as elegant.
Consider $(n+1)^3 + (n-1)^3 = 2n^3 + 6n$
Thus,
$6n = (n+1)^3 + (n-1)^3 + (-n)^3 + (-n)^3$
Effectively proving the theorem and also giving the required four numbers. The professor also made a remark that a proof is due to Ramanujan. I recently found this scribbled in my notebook and have since not been able to find any reference to this on the web. As far as I know, Kennigel's biography, 'The Man Who Knew the Infinity' does not mention it. There are tons of references to Taxicab numbers, Sum of four square proofs etc..
Does anyone know of any reference to the theorem and the proof? Is it a part of a more general theory?
Indeed, it was who discovered the sum of four squares, it was G.Xylander that tried to solve what Diophantus had pointed much earlier. Also, it was discovered that, by Fermat that every number is the sum of four squares. The final proof came with the Lagrange's Four-Square Theorem that follows from Euler's Four-Square Identity.
And the proof uses quarternions and can be found here: https://en.wikipedia.org/wiki/Lagrange%27s_four-square_theorem
However, the sum of cubes for a multiple of 6 was generalized by Edward Waring, Hibert Theorem (which as to find $g(k)$, this is, for every ${\displaystyle k}$ , let ${\displaystyle g(k)}$ denote the minimum number ${\displaystyle s}$ of ${\displaystyle k}$th powers of naturals needed to represent all positive integers). And the source is: https://en.wikipedia.org/wiki/Waring%27s_problem