Starting from Diophantus’ classical question on expressing integers as sums of squares and Lagrange’s four-square theorem, we are led to Waring’s problem, which asks whether every natural number can be written as a sum of finitely many k-th powers. Hilbert showed that this is always possible, motivating the study of how efficiently integers can be represented by powers.
A very natural mathematical question is how integers can be constructed using simple building blocks such as squares and cubes. While squares exhibit a remarkably regular behavior, cubes lead to far more subtle and less understood phenomena, where even basic-looking questions become nontrivial and rich in structure.
In this talk, we focus on representations of integers as sums of cubes, with special emphasis on expressions as sums of two rational cubes. We explain how such representations naturally lead to classical Diophantine equations of the form
a³ + b³ =n,
and how questions about existence and structure of solutions can be studied through this reformulation.
We also briefly mention related problems involving sums of three, four, and more cubes, highlighting the contrast between simple statements and deep arithmetic constraints. The main goal is to illustrate how elementary questions about cubes naturally translate into Diophantine problems, where modern ideas provide tools to understand them.
We look forward to your active participation and an engaging discussion.
