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

TL;DR

This paper provides computational evidence that the specific family of quintic polynomials related to the second cuboid problem do not factor into quadratic and cubic polynomials over the rationals, suggesting such factorizations are unlikely.

Contribution

The authors analyze a parametric family of quintic polynomials derived from the second cuboid problem and provide computational evidence against their quadratic-cubic factorizations.

Findings

No rational points with s>0 and s≠1 found up to height 10^9

The polynomial P_s(x) likely admits no quadratic factor over Q for s>0, s≠1

Conditional on the completeness of the rational point search, quadratic-cubic factorizations are excluded

Abstract

Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\neq q>0$. Writing $Q_{p,q}(t)$ as a quintic in $t^{2}$ produces an associated monic quintic polynomial. After the weighted normalization $r=p/q$ and $s=r^{2}$ we obtain a one-parameter family $P_s(x)\in\mathbb{Q}[x]$ such that \[ Q_{p,q}(t)=q^{20}\,P_s\!\left(\frac{t^{2}}{q^{4}}\right)\qquad\text{with}\qquad s=\left(\frac{p}{q}\right)^{2}. \] Assuming a quadratic divisor $x^{2}+ax+b$ with $a,b\in\mathbb{Q}$, we reduce divisibility of $P_s(x)$ to the vanishing of an explicit remainder \[ R(x)=R_{1}(s,a,b)\,x+R_{0}(s,a,b). \] A key structural observation is that $R_1$ and $R_0$ are quadratic in $b$ and that, on the equation $R_1=0$, the second condition becomes linear in $b$. This yields a one-direction elimination to a plane obstruction curve $F(s,a)=0$ with $F\in\mathbb{Z}[s,a]$, without any lifting-back issues: when the linear coefficient is nonzero, the parameter $b$ is forced to be the rational value $b=C/L$. We isolate the degenerate locus $L=C=0$ and show it produces only $s=\pm 1$ (hence only $s=1$ in the cuboid domain $s>0$). Let $\overline{C}\subset\mathbb{P}^{2}$ be the projective closure of $F(s,a)=0$. Using Magma we perform a height-bounded search for rational points on $\overline{C}$. With bound $H=10^{9}$, the search returns $8$ rational points, whose affine part has $s\in\{-1,0,1\}$. In particular, no affine rational point with $s>0$ and $s\neq 1$ is found up to this bound. This provides strong computational evidence that for rational $s>0$, $s\neq 1$, the quintic $P_s(x)$ admits no quadratic factor over $\mathbb{Q}$ (equivalently, no $2+3$ (quadratic-cubic) factorization over $\mathbb{Q}$), and yields a conditional exclusion assuming completeness of the rational-point enumeration on $\overline{C}$.