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Computational Evidence Against Quadratic-Cubic Factorization for the Second Cuboid Quintic

This paper provides strong computational evidence that Sharipov's second cuboid quintic polynomial admits no quadratic-cubic factorization over the rationals for any rational parameter $s>0$ (except $s=1$ ), based on a structural reduction to an obstruction curve and an exhaustive height-bounded search for rational points up to $10^9$ .

Original authors: Valery Asiryan, Randall L. Rathbun

Published 2026-01-23

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CC BY 4.0

Original authors: Valery Asiryan, Randall L. Rathbun

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 are trying to build a perfect rectangular box (a "cuboid") where every single measurement is a whole number. You want the length, width, and height to be integers. But you also want the diagonal lines across every face and the diagonal line cutting through the very center of the box to be whole numbers too.

This is the Perfect Cuboid Problem. It's a famous puzzle in math that has stumped people for centuries. No one has ever found such a box, and no one has proven it's impossible to build one.

This paper is like a team of detectives (Valery Asiryan and Randall Rathbun) using a super-powered computer to check one specific way someone might try to build this box. They aren't building the box; they are checking the blueprints to see if a specific type of blueprint is even possible.

Here is how they did it, explained simply:

1. The Blueprint (The Polynomial)

Mathematicians have turned the rules of the box into a giant algebraic equation (a polynomial). Think of this equation as a complex recipe. If you can find a specific ingredient (a number) that makes the recipe work, you might find your box.

The authors focused on a specific version of this recipe, which they call a "quintic" (a 5th-degree equation). They wanted to know: Can this recipe be broken down into smaller, simpler recipes? specifically, can it be split into a "quadratic" (2nd-degree) part and a "cubic" (3rd-degree) part?

If the answer is yes, it means the box might be buildable using that specific method. If the answer is no, that particular path to building the box is a dead end.

2. The Detective Work (The Remainder)

To see if the recipe can be split, the authors tried to divide the big equation by a generic "quadratic" piece. In math, when you divide things, you usually get a result and a "remainder" (what's left over).

The Goal: For the split to work, the remainder must be exactly zero.
The Trick: The authors discovered that the "remainder" wasn't just a messy jumble. It had a special structure. It depended on two variables, let's call them $a$ and $b$ .
The Breakthrough: They realized that if they set one part of the remainder to zero, the other part became a simple, straight-line equation for $b$ . This allowed them to eliminate $b$ entirely and focus only on $a$ and another variable $s$ (which represents the shape of the box).

3. The Obstacle Course (The Curve)

By eliminating $b$ , they created a new map called an "obstruction curve." Think of this as a fence drawn on a piece of paper.

If you can find a point on this fence where both coordinates are whole numbers (or simple fractions), then the box might exist.
If the fence has no such points, then that specific way of building the box is impossible.

This fence is a very complex, twisted shape (mathematically, a curve with a high "genus," meaning it has many holes and twists). Finding whole-number points on such shapes is notoriously difficult, like finding a specific grain of sand on a massive beach.

4. The Computer Search (The Height-Bounded Hunt)

Since the curve is so complex, the authors couldn't check every single point. Instead, they used a powerful computer program called Magma to perform a "height-bounded search."

The Analogy: Imagine searching for a specific house in a city. You can't check every house in the world, so you decide to only look at houses within a 1-billion-mile radius of the center.
The Search: They told the computer to look for all "rational" points (simple fractions) on their curve within a massive but finite range (a "height" of $10^9$ ).

5. The Results

The computer finished the search and found only 8 points on the entire curve.

Most of these points were "at infinity" (mathematical edge cases that don't represent real boxes).
The few points that were "real" (on the flat part of the map) only had coordinates where the shape variable $s$ was 0, 1, or -1.
The Catch: In the world of the perfect cuboid, the shape variable $s$ must be a positive number not equal to 1.

The Conclusion:
The computer found zero points on the curve that fit the rules for a real, non-trivial box.

What This Means

The authors state that this provides strong computational evidence that this specific way of building a perfect cuboid (splitting the equation into a quadratic and a cubic) is impossible.

They did not prove the Perfect Cuboid Problem is impossible forever.
They did not prove that no other method exists.
They did prove that for this specific mathematical path, the "fence" has no valid stepping stones for anyone to walk on, at least within the massive range the computer checked.

In short: They checked a very specific, complicated door to see if it leads to the Perfect Cuboid. Their computer search suggests the door is locked tight, and there are no keys (rational points) hidden in the area they searched.

Technical Summary: Computational Evidence Against Quadratic–Cubic Factorization for the Second Cuboid Quintic

Problem Statement
The paper addresses a specific algebraic aspect of the Perfect Cuboid problem, which seeks a rectangular box with integer edges where all face and space diagonals are integers. Following the framework established by R. A. Sharipov, the problem reduces to proving the irreducibility of a family of degree-10 even polynomials, $Q_{p,q}(t)$ , over the integers for coprime parameters $p \neq q > 0$ .

The authors focus on the "quadratic–cubic" (2+3) factorization case. By normalizing the variables, the degree-10 polynomial $Q_{p,q}(t)$ is transformed into a one-parameter family of monic quintic polynomials, $P_s(x)$ , where $s = (p/q)^2$ . The central question is whether $P_s(x)$ admits a factorization into a quadratic and a cubic polynomial over $\mathbb{Q}$ for any rational $s > 0$ where $s \neq 1$ . A positive answer would imply the existence of an even quartic factor in the original polynomial, potentially obstructing the irreducibility conjecture.

Methodology
The authors employ a direct algebraic approach combined with computational number theory:

Algebraic Reduction:
- They consider a generic monic quadratic divisor $D(x) = x^2 + ax + b$ with $a, b \in \mathbb{Q}$ .
- Using Euclidean division, they compute the remainder $R(x) = R_1(s, a, b)x + R_0(s, a, b)$ when $P_s(x)$ is divided by $D(x)$ .
- A key structural observation is that both $R_1$ and $R_0$ are quadratic in the parameter $b$ .
- By imposing the condition $R_1 = 0$ , the authors show that the second condition $R_0 = 0$ becomes linear in $b$ . This allows for the elimination of $b$ to derive a single polynomial equation $F(s, a) = 0$ in the $(s, a)$ -plane, termed the "obstruction curve."
Case Analysis:
- Non-degenerate Case ( $L \neq 0$ ): When the linear coefficient $L(s, a)$ is non-zero, $b$ is uniquely determined as $b = C(s, a)/L(s, a)$ . Substituting this back yields the obstruction polynomial $F(s, a) = 0$ .
- Degenerate Case ( $L = C = 0$ ): The authors separately analyze the locus where the linearization fails. They prove via resultants that the only rational solutions to $L(s, a) = C(s, a) = 0$ are $(s, a) = (1, 2)$ and $(-1, 2)$ . Since the problem domain requires $s > 0$ and $s \neq 1$ , this degenerate locus yields no valid solutions.
Computational Search:
- The obstruction polynomial $F(s, a)$ defines a projective plane curve $C$ of degree 17 with geometric genus 7.
- Determining all rational points on such curves is a difficult Diophantine problem. Instead of attempting a complete theoretical determination (e.g., via Chabauty–Coleman methods), the authors perform a height-bounded search using the computer algebra system Magma.
- They search for rational points on the projective closure of $C$ with a height bound of $H = 10^9$ .

Key Results

Structure of the Obstruction: The authors explicitly derive the polynomials $u, v, m, n, L, C$ and the obstruction polynomial $F(s, a)$ . They verify that $F(s, a)$ has degree 16 in $s$ and degree 10 in $a$ .
Degenerate Locus: The system $L=C=0$ is shown to have only the rational solutions $(1, 2)$ and $(-1, 2)$ . Consequently, for the relevant domain $s \in \mathbb{Q}_{>0}, s \neq 1$ , the degenerate case is empty.
Rational Points Search: The Magma search with bound $H = 10^9$ returns exactly 8 rational points on the curve $C$ . The affine part of these points corresponds to $s \in \{-1, 0, 1\}$ .
Exclusion of Factorization: Crucially, no affine rational point with $s > 0$ and $s \neq 1$ is found within the search bound.

Claims and Significance
The paper claims strong computational evidence rather than a definitive unconditional proof. The significance of the work is framed as follows:

Conditional Exclusion: The authors establish a conditional theorem: Assuming the list of 8 rational points found is complete (Conjecture 1), then for any rational $s > 0$ with $s \neq 1$ , the quintic $P_s(x)$ admits no quadratic factor over $\mathbb{Q}$ .
Implication for the Cuboid Problem: Under this assumption, the second cuboid polynomial $Q_{p,q}(t)$ has no even quartic factor over $\mathbb{Q}$ for coprime $p \neq q$ .
Modesty of Claims: The authors explicitly acknowledge that a complete determination of the rational points $C(\mathbb{Q})$ is required for an unconditional theorem. They note that the curve has geometric genus 7, making the enumeration of rational points a subtle problem where finiteness is guaranteed by Faltings' theorem, but explicit determination typically requires advanced techniques beyond a simple height search.

In summary, the paper reduces the factorization problem to finding rational points on a specific high-genus curve and provides extensive computational data suggesting that no such points exist in the relevant domain, thereby supporting the conjecture of irreducibility for the second cuboid polynomial.