Given a positive integer $n$, consider expressions of the form
$$ n = x \cdot y + z $$
where $x, y, z \in \mathbb{N}$, with constraints $0 < x, y \leq n$ and $0 \leq z < x$. Such a triple $(x, y, z)$ is called a Euclidean decomposition of $n$. These decompositions arise naturally from the division algorithm and can be interpreted geometrically as lattice-based representations of $n$ in the first quadrant.
The central question addressed in this note is:
Among all such Euclidean decompositions of $n$, which one minimizes the quantity $x + y + z$?
This total sum can be viewed as a proxy for the simplicity or efficiency of the decomposition. While the constraints allow many representations for each $n$, the goal is to identify the one where the combined cost of $x$, $y$, and $z$ is as small as possible.
The note develops this idea formally, proving several lemmas that characterize the behavior of such minimal decompositions. These include:
- Bounding $x$ and $y$ in terms of $n$
- Showing structural patterns in optimal decompositions
- Reducing the search space for efficient computation
Combining these observations leads to an algorithm with time complexity $O(n^{3/8})$, significantly improving over a naive brute-force approach.