Abstract: We present a simple geometric method called "cube extension" to compare the Diophantine equations x³ + y³ = z³ and w³ = x³ + y³ + z³. Starting from the identity (n+k) ³ - n³ = 3n² k + 3n k² + k³, we interpret the right‑hand side as the volume of three slabs added to a cube of side n to obtain a cube of side n+k. We prove that this added volume can never be a perfect cube (which gives an elementary proof of the n=3 case of Fermat's Last Theorem), but it can often be expressed as a sum of two cubes, leading to infinitely many integer solutions of w³ = x³ + y³ + z³. Explicit primitive solutions and parametric families are provided, including a step‑by‑step numerical illustration of growing a cube from side 40 to side 50. The method is constructive, visual, and suitable for a wide audience.
Emma Helmdach (Thu,) studied this question.