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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?

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    $\begingroup$ You might want to look at mathworld.wolfram.com/CubicNumber.html it mentions this identity along with some more about writing whole numbers as the sum of 4 signed cubes. $\endgroup$ Jun 27 '13 at 16:04
  • $\begingroup$ Cool, I missed the Wolfram page. The page does provide a general set of theorems for representing numbers as sums of cubes. The identity in question does appear in on the page, so does the Taxicab number. But it does not mention the source of the identity. Maybe it did not originate from Ramanujan and my notes were erroneous. $\endgroup$
    – Sudeep
    Jun 27 '13 at 16:32
  • $\begingroup$ the book Hardy wright theory of numbers has the equation you stated.author is probably Mordell and Hammond. $\endgroup$
    – user242371
    May 21 '15 at 10:20
  • $\begingroup$ It follows that all primes $\ge5$ are the sum of five cubes. $\endgroup$
    – Lucian
    Sep 20 '16 at 12:24
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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

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