By applying an existing characterization for a positive integer to be represented as a sum of two cubes of positive integers, we construct an elementary proof of Ramanujan's famous result—namely, that the number 1729 is the smallest positive integer represented as a sum of two cubes in two different ways. Similarly, by applying an existing characterization for a positive integer to be represented as a difference of cubes of two positive integers, we also apply this characterization to construct the generating function for a sequence of integer ordered pairs (a_n, b_n) ≠ (a'_n, b'_n) satisfying a_n³ + b_n³ = a'³_n + b'³_n, which are distinct from Ramanujan's "near integer" solutions to Fermat's equation—namely, those satisfying a_n³ + b_n³ = a_n³ + (–1)ⁿ.