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There are infinitely many Hilbert cubes of dimension 3 in the set of squares

This paper resolves the question of whether the maximal dimension of Hilbert cubes in the set of squares is bounded by proving that there exist infinitely many 3-dimensional Hilbert cubes within the squares, with their defining parameters forming a dense set of ratios in the positive real numbers.

Original authors: Andrew Bremner, Christian Elsholtz, Maciej Ulas

Published 2026-04-08

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Andrew Bremner, Christian Elsholtz, Maciej Ulas

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a giant, infinite shelf filled with perfect square numbers: 1, 4, 9, 16, 25, 36, and so on. Now, imagine you want to build a special structure using these numbers.

This structure is called a Hilbert Cube. Think of it not as a 3D box made of wood, but as a mathematical "recipe" for generating numbers.

The Recipe Analogy

To make a Hilbert Cube of dimension 3 (a 3D cube), you need four ingredients: a starting number ( $a_0$ ) and three "step" numbers ( $a_1, a_2, a_3$ ).

The rule is simple: You start at $a_0$ . Then, you can choose to take zero steps or one step of size $a_1$ , zero or one step of size $a_2$ , and zero or one step of size $a_3$ .

If you mix and match these choices, you get $2^3 = 8$ different numbers.

The Goal: All 8 of these resulting numbers must be perfect squares.

For example, if you pick the right ingredients, you might get a set like $\{100, 2500, 4900, 6400, \dots\}$ where every single number is a perfect square.

The Big Question

For a long time, mathematicians wondered: How big can these cubes get?

Can we find a 4D cube (16 numbers) where every number is a square?
Can we find a 100D cube?
Or is there a "ceiling" that stops us from going higher?

Previous research suggested that if you look at squares up to a very large number $N$ , the size of the cube can't grow too fast. But nobody knew if there was an absolute limit (like "you can never go higher than dimension 3") or if you could keep building bigger and bigger cubes forever.

What This Paper Discovered

The authors (Andrew Bremner, Christian Elsholtz, and Maciej Ulas) decided to stop guessing and start building.

1. They proved there are infinitely many 3D cubes.
They didn't just find one or two; they found a "factory" (a mathematical formula) that can produce an infinite number of these 3D cubes.

The Analogy: Imagine you have a magic machine. You put in a number, and it spits out a perfect 3D cube made of squares. They proved this machine never runs out of fuel.
The Count: They showed that if you look at all squares up to a number $N$ , you will find roughly $N^{1/8}$ of these cubes. That's a lot! It means they are not rare anomalies; they are common enough to be found systematically.

2. They found "twin" cubes.
They discovered that you can have two different cubes that share the first three ingredients ( $a_0, a_1, a_2$ ) but have a different fourth ingredient ( $a_3$ ).

The Analogy: It's like having two different cars that share the same engine, chassis, and wheels, but have different paint jobs. This was a surprising discovery that helps mathematicians understand the "shape" of these numbers.

3. They explored the "Magic Square" connection.
One of the open questions in math is whether you can make a "Magic Square of Squares" (a 3x3 grid where every row, column, and diagonal is a square number). The authors showed that finding a specific type of 4D Hilbert cube is mathematically equivalent to solving this Magic Square puzzle. While they didn't solve the Magic Square yet, they got very close, finding "almost" solutions where 13 out of 16 numbers were squares.

4. The Density Result.
They proved that the ratios between the ingredients of these cubes can be almost anything you want.

The Analogy: If you have a bag of these cubes, you can pick one where the second ingredient is half the size of the first, or one where it's a million times bigger. The possibilities are "dense," meaning they cover the entire spectrum of numbers without gaps.

The Bottom Line

Before this paper, we knew 3D cubes of squares existed, but we didn't know how many there were or if we could keep finding them forever.

The authors proved:

Yes, there are infinitely many 3D cubes of squares.
We can generate them using specific formulas.
We still haven't found a 4D cube (16 numbers), and it might be impossible, but we are now much better equipped to search for it or prove it doesn't exist.

In short, they turned a mysterious, rare mathematical curiosity into a well-understood, abundant family of numbers, while still leaving the door open for the ultimate challenge: the 4D cube.

1. Problem Statement

The paper investigates the existence and density of Hilbert cubes within the set of perfect squares, $S = \{n^2 : n \in \mathbb{N}\}$ .

Definition: A Hilbert cube of dimension $d$ is a set of integers defined as:
$H(a_0; a_1, \dots, a_d) = a_0 + \{0, a_1\} + \dots + \{0, a_d\} = \left\{ a_0 + \sum_{i=1}^d \epsilon_i a_i : \epsilon_i \in \{0, 1\} \right\}$
The set contains $2^d$ elements (some of which may coincide if $a_i$ are not distinct).
The Core Question: Do there exist Hilbert cubes of arbitrarily high dimension contained entirely within the set of squares?
- It is known that infinite Hilbert cubes do not exist in $S$ .
- Brown, Erdős, and Freedman asked if the maximal dimension $d$ is absolutely bounded.
- Previous work (Dietmann and Elsholtz) established that if a cube is contained in $S \cap [0, N]$ , then $d \le O(\log \log N)$ .
- The Bombieri-Lang conjecture implies that $d$ is absolutely bounded (likely $d \le 3$ or $4$).
Specific Focus: The authors focus on dimension $d=3$ . While examples of $d=3$ cubes were known, it was unclear if there were infinitely many such cubes, nor was there a quantitative lower bound on their number. The paper also addresses the existence of $d=4$ cubes (finding none) and the density of ratios between cube elements.

2. Methodology

The authors employ a combination of computational number theory, algebraic geometry (specifically the theory of elliptic curves), and parametric constructions.

A. Computational Search and Algorithm

Algorithm 1: They developed an efficient algorithm to find Hilbert cubes $H(a_0; a_1, a_2, a_3) \subset S$ $H (a_{0}; a_{1}, a_{2}, a_{3}) \subset S$ with $a_0, a_1 \le N$ $a_{0}, a_{1} \leq N$ .
- The algorithm iterates through squares $a_0 = p^2$ and $a_0+a_1 = q^2$ .
- It utilizes the divisor structure of $a_1$ to solve for $a_2$ and $a_3$ by checking if specific quadratic forms are squares.
- Complexity: The algorithm runs in time $O_\epsilon(N^{1+\epsilon})$ .
Data Generation: They computed all reduced Hilbert cubes of dimension 3 with $\max(a_i) \le 2^{18}$ , finding 4644 such cubes. None could be extended to dimension 4.

B. Parametric Construction via Elliptic Curves

To prove the existence of infinitely many cubes, the authors moved from discrete search to continuous parameterization:

System of Equations: They formulated the condition that all $2^3=8$ elements of the cube are squares as a system of Diophantine equations.
Reduction to Elliptic Curves: By fixing certain variables and utilizing the structure of the equations, they reduced the problem to finding rational points on specific elliptic curves.
- They constructed a curve $E$ in projective space.
- They identified points of infinite order on this curve (e.g., $Q_1, Q_2$ ).
Parametrization: By pulling back multiples of these points (e.g., $2P_0$ $2 P_{0}$ ), they derived explicit polynomial parametrizations for $(a_0, a_1, a_2, a_3)$ $(a_{0}, a_{1}, a_{2}, a_{3})$ in terms of parameters (e.g., $c, d, G, H$ $c, d, G, H$ or a single parameter $t$ $t$ ).
- This allows the generation of an infinite family of solutions.

C. Density Analysis

To prove the density of ratios $a_i/a_j$ , they analyzed the rational functions derived from their parametric families. They showed that these functions map rational intervals to the entire set of positive real numbers $\mathbb{R}^+$ , proving density in the Euclidean topology.

3. Key Contributions and Results

A. Existence of Infinitely Many Dimension-3 Cubes

Theorem 1.5: The number of reduced Hilbert cubes of dimension 3 with entries $\le N$ , denoted $H_3(N)$ , satisfies:
$H_3(N) \gg N^{1/8}$
This proves there are infinitely many such cubes.
Theorem 1.6: There are infinitely many reduced Hilbert cubes where $a_1 = a_2 < a_3$ (and symmetrically $a_1 < a_2 = a_3$ ). The count for these specific cases is at least $cN^{1/10}$ .

B. Density of Ratios

Corollary 1.8: For any pair of indices $i < j$ , the set of ratios $\{a_i / a_j\}$ derived from all such Hilbert cubes is dense in the set of positive real numbers $\mathbb{R}^+$ . This implies that the relative sizes of the components of these cubes can approximate any positive real ratio.

C. Non-Existence of Dimension-4 Cubes (Empirical & Partial)

Despite extensive computational searches and theoretical attempts, the authors found no Hilbert cubes of dimension 4 in the set of squares.
They constructed "pseudo-4-cubes" where 13 out of the 16 required sums are squares, but the remaining 3 are not.
They note that finding a true $d=4$ cube would likely require a breakthrough, as it is conjectured (via Bombieri-Lang) that no such cubes exist.

D. Fixed Base $a_0$

Conjecture 7.1: For every square $a_0 = n^2$ , there exists a reduced Hilbert cube of dimension 3.
Computational Evidence: They verified this for $n \in \{1, \dots, 100\}$ , finding solutions for all but four cases ( $n=42, 44, 56, 80$ ).

E. Quantitative Growth

Using numerical fitting on their data ( $N \le 10^7$ ), they suggest the growth rate of $H_3(N)$ is approximately $N^{0.62}$ , which is significantly higher than the proven lower bound of $N^{1/8}$ .

4. Significance

Resolution of Open Questions: The paper definitively answers the question of whether infinitely many dimension-3 Hilbert cubes exist in the squares (Yes).
Quantitative Bounds: It provides the first explicit lower bound ( $N^{1/8}$ ) for the number of such cubes, moving beyond qualitative existence.
Methodological Advancement: The work demonstrates a powerful application of elliptic curves to construct families of solutions for high-dimensional additive problems in number theory.
Insight into Higher Dimensions: While it confirms the abundance of $d=3$ cubes, the failure to find $d=4$ cubes (and the theoretical barriers) reinforces the hypothesis that the maximal dimension in squares is likely 3, aligning with the Bombieri-Lang conjecture.
Generalization: The authors briefly discuss generalizing these results to other quadratic polynomials (e.g., triangular numbers), showing that dimension 3 is achievable there as well, and posing the open problem of finding dimension 4 in other quadratic sets.

Summary Conclusion

The paper establishes that the set of squares is "rich" enough to contain infinitely many Hilbert cubes of dimension 3, with a specific density of ratios between their elements. However, it suggests a hard barrier at dimension 4, supporting the conjecture that the maximal dimension of a Hilbert cube in the squares is 3. The results are achieved through a sophisticated blend of algorithmic search and the construction of parametric families via elliptic curves.