A Closed-Form Symbolic Generator: Aⁿ + Bⁿ = Cⁿ + Dⁿ, for n = 2, 3

We present a unified framework for constructing integer solutions to A^n + B^n = C^n + D^n for n=2, 3. For n=2, we derive explicit formulas for any solutions via differences of squares. For n=3, we introduce general formulas that include the Hardy-Ramanujan number 1729 for instance, we also construct a symbolic generator that produces infinitely many integer solutions to the Diophantine equation A³ + B³ = C³ + D³. While the resulting formulas for A, B, C, D from the symbolic generator developed do not span every single number expressible as a sum of two positive cubes in at least two distinct ways, our method provides a closed-form, algebraic parametrization in terms of a single variable, expressing each term as a radical-exponential function of an integer parameter c₁. The generator leverages nested radicals and exponents of algebraic numbers, α, β derived from the recurrence structure of the Diophantine constraint. This work represents the first symbolic, recursive generator of its kind and offers a pathway toward approaching higher powers of this problem from a different lens. These methods exploit structural links between binomial expansions and Diophantine constraints, offering a foundation for extensions to higher powers.