# Proving that any rational number can be represented as the sum of the each cube of three rational numbers

I found the following question in a book:

Prove that any integer can be represented as the sum of the each cube of five integers.

The answer : $$n=n^3+\left[-\frac{(n-1)n(n+1)}{6}-1\right]^3+\left[-\frac{(n-1)n(n+1)}{6}-1\right]^3+\left[\frac{(n-1)n(n+1)}{6}\right]^3+\left[\frac{(n-1)n(n+1)}{6}\right]^3.$$

This book says, "It is known that any rational number can be represented as the sum of the each cube of three rational numbers" without any additional information. I've tried to prove this, but I'm facing difficulty. Then, here is my question.

Question : Could you show me how to prove that any rational number can be represented as the sum of the each cube of three rational numbers.

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I believe this is known as Ryley's Theorem –  Mufasa Sep 15 '13 at 14:36

Theorem (Reyley 1825) Every rational number $R$ can be represented as the sum of three rational cubes: $$R=x^3+y^3+z^3.$$ A short proof can be found in the artcle of Richmond (1930) here: journals.cambridge.org/article_S0013091500007604. See also here: http://www.mast.queensu.ca/~reidl/ColemanEllis/CE6.html.

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Thank you for your answer! –  mathlove Sep 15 '13 at 14:50