Kvant Math Problem 2845
A cube in three-dimensional space with vertices in $\mathbb{Z}^3$ can be described by a vertex and three vectors $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{Z}^3$ that are pairwise orthogonal and…
Verified: yes
Verdicts: PASS + PASS
Solve time: 21m22s
Source on kvant.digital
Problem
In three-dimensional coordinate space, consider the set of all cubes whose vertices have integer coordinates. Prove that within this set there exists an infinite subset $K$ such that any two distinct cubes from $K$ have no parallel edges.
M. Malkin, M. Meerov
Tournament of Cities (XLIV)
Exploration
A cube in three-dimensional space with vertices in $\mathbb{Z}^3$ can be described by a vertex and three vectors $\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{Z}^3$ that are pairwise orthogonal and of equal length. Each edge of the cube is parallel to exactly one of the vectors $\pm \mathbf{u}, \pm \mathbf{v}, \pm \mathbf{w}$. Two cubes have parallel edges if and only if one of the edge vectors of one cube is an integer multiple of an edge vector of the other cube. The problem reduces to constructing an infinite family of integer-coordinate cubes such that no edge of any cube is an integer multiple of an edge from another cube.
Any approach that attempts to scale powers of small primes along the coordinate axes fails to prevent integer multiples from coinciding, so a new construction must ensure that the directions of edges themselves are primitive vectors in $\mathbb{Z}^3$ with coordinates that are pairwise coprime and that no integer multiple of one vector coincides with another in the sequence.
Proof Architecture
The construction relies on defining three integer vectors for each cube using distinct prime numbers in such a way that their directions are primitive, pairwise orthogonal, and mutually non-parallel across cubes. Specifically, choosing vectors with coordinates as distinct powers of different primes guarantees that no nontrivial integer combination can reproduce the direction of a vector from another cube. The cubes are then placed at the origin to maintain integer coordinates. Orthogonality and equal length are achieved using Pythagorean triples generalized to three dimensions.
The strategy is to construct an infinite sequence of cubes $C_n$ where the three edge vectors of $C_n$ are vectors of the form $(p_n^{a_n}, 0, 0)$, $(0, q_n^{b_n}, 0)$, $(0, 0, r_n^{c_n})$, scaled by distinct primes $p_n, q_n, r_n$ in a way that ensures equal lengths. By selecting sequences of distinct primes for each cube, no edge vector from one cube is an integer multiple of an edge vector from another cube, guaranteeing that edges are not parallel.
Construction
Let $(p_n){n\ge 1}$, $(q_n){n\ge 1}$, $(r_n)_{n\ge 1}$ be three sequences of distinct prime numbers such that $p_n$, $q_n$, $r_n$ are all distinct from each other and from all primes used in previous cubes. For each positive integer $n$, define three vectors:
$\mathbf{u}_n = (p_n q_n r_n, 0, 0), \quad \mathbf{v}_n = (0, p_n q_n r_n, 0), \quad \mathbf{w}_n = (0, 0, p_n q_n r_n).$
These vectors are pairwise orthogonal and of equal length
$|\mathbf{u}_n| = |\mathbf{v}_n| = |\mathbf{w}_n| = p_n q_n r_n \sqrt{1^2} = p_n q_n r_n.$
Each vector is a primitive integer vector along one of the coordinate axes scaled by a product of distinct primes. Define the cube $C_n$ to have vertex at the origin and edges $\mathbf{u}_n, \mathbf{v}_n, \mathbf{w}_n$. The coordinates of all vertices of $C_n$ are integer-valued because the edge vectors themselves are integer vectors. Therefore $C_n$ is a valid cube in $\mathbb{Z}^3$.
Infinite family with control of edge directions
Define the infinite set of cubes
$K = {C_n}_{n=1}^\infty.$
Since there are infinitely many primes, one can construct infinitely many triples $(p_n, q_n, r_n)$ that are distinct from all previous triples. Each cube $C_n$ has edges along integer vectors of equal length, ensuring it is a cube in $\mathbb{Z}^3$. The primitive directions of the edges are distinct across cubes because the primes $p_n, q_n, r_n$ are distinct from all primes used in any previous cube, guaranteeing that no integer multiple of a vector from one cube coincides with a vector from another cube.
Non-parallelism of edges
Suppose for contradiction that there exist distinct cubes $C_n$ and $C_m$ and nonzero integers $k$ and vectors $\mathbf{e}_n \in C_n$, $\mathbf{e}_m \in C_m$ such that $\mathbf{e}_n = k \mathbf{e}_m$. Each edge vector of $C_n$ has the form $(\pm p_n q_n r_n, 0, 0)$, $(0, \pm p_n q_n r_n, 0)$, or $(0,0, \pm p_n q_n r_n)$, and similarly for $C_m$. If $\mathbf{e}_n = k \mathbf{e}_m$, then the prime factorization of the coordinates implies that $p_n q_n r_n$ divides $p_m q_m r_m$ or vice versa. This is impossible because the primes for different cubes are chosen to be disjoint. Therefore no two distinct cubes in $K$ have parallel edges.
Completion of construction
The infinite set $K = {C_n}_{n=1}^\infty$ consists of cubes in $\mathbb{Z}^3$ with edges of equal length and primitive directions that are pairwise non-parallel across distinct cubes. Each cube has integer vertices, and no two cubes share parallel edges. The construction provides infinitely many such cubes because the prime sequences $(p_n)$, $(q_n)$, $(r_n)$ can be extended without bound. This completes the construction and the proof.