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

Abstract

Let $Q_{p,q}(t)\in \mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\ne q>0$. 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 $s\ne 1$ is found up to this bound. This provides strong computational evidence that for rational $s>0$, $s\ne 1$, 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 $\overline{C}$.

Partial progress on the irreducibility of the second cuboid polynomial (Sharipov's second conjecture): computational evidence against 2+3 factorization for the associated quintic