Telescoping Ramanujan Identities and Massive Decompositions of Cubes into Sums of Cubes

We present a deterministic algebraic method to decompose the cube of an integer into a sum of an arbitrarily large number of positive integer cubes. The method exploits a hidden telescoping property in a known parameterized system of Diophantine equations by Ramanujan for the sum of three cubes. By evaluating the polynomial parameters at sequential integer points, the intermediate terms cancel out in a telescoping series. This leads to an exact representation of W^3 as a sum of 2m+1 positive cubes, generated in O(1) space and O(m) time. We explicitly demonstrate the algorithmic construction and showcase decompositions consisting of up to 105,663 terms.