On the Irreducibility of the Cuboid Polynomial P_a,u(t)

2025/10/09 by Valery Asiryan, Asiryan, Valery · 1 citation
Computer Science · Mathematics · #11D72 #11S05 #11Y05 #12E05 #Algebraic Geometry and Number Theory #FOS: Mathematics #General Mathematics (math.GM) #Mathematical Dynamics and Fractals #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2510.07643

openalex publication_date 2025/10/09 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the even monic degree-8 cuboid polynomial P_a,u(t) with coprime integers a≠ u>0. We prove irreducibility over 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 ℤ[t].

Citations

• A note on a perfect Euler cuboid
• Unsolved problems in number theory

Cited by

• Irreducibility of the Cuboid Polynomial P_a,u(t) via a Rank-Zero Elliptic Curve

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