On the Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$
Valery Asiryan
TL;DR
The paper proves the irreducibility over $\mathbb{Z}$ of the even monic cuboid polynomial $P_{a,u}(t)$ for coprime $a\neq u>0$ by exhaustively excluding all degree-8 factorizations. The core strategy splits into ruling out a $4{+}4$ factorization via a Diophantine condition $(X^{2}-8\Delta^{2})(X^{2}-9\Delta^{2})=4a^{2}u^{2}X^{2}$ and applying $2$- and $3$-adic analyses, reinforced by a gcd lemma $\gcd(X,\Delta)=1$; it then eliminates a $2{+}6$ factorization using an even-quadratic divisor criterion and a discriminant obstruction. The remaining patterns $2{+}2{+}4$, $2{+}2{+}2{+}2$, and $3{+}3{+}2$ are shown to regroup to $2{+}6$ and are thus impossible. A residual elliptic-curve analysis is invoked to handle a delicate residual case, ensuring the no-solution conclusion. Overall, the paper establishes full irreducibility of $P_{a,u}(t)$, contributing a rigorous constraint toward the algebraic structure surrounding the cuboid polynomials relevant to Euler-type cuboids.
Abstract
In this paper we consider the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ with coprime integers $a\neq u>0$. We prove irreducibility over $\mathbb{Z}$ by excluding all degree-8 splittings. First, any putative $4{+}4$ factorization is shown to force a specific Diophantine constraint that has no integer solutions, via a short $2$- and $3$-adic analysis. Second, we exclude every $2{+}6$ factorization using an exact divisor criterion together with a discriminant obstruction. Finally, after ruling out $2{+}6$, the patterns $2{+}2{+}4$, $2{+}2{+}2{+}2$, and $3{+}3{+}2$ regroup trivially to $2{+}6$ and are therefore impossible. Consequently, $P_{a,u}(t)$ admits no nontrivial factorization in $\mathbb{Z}[t]$.