2024/07/13 by Takumi Yoshida, Yoshida, Takumi
Computer Science · Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2407.09825
openalex publication_date 2024/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A rational face cuboid is a cuboid that all of edges, two of three face diagonals and space diagonal have rational lengths. E1,s: y2=x(x-(2s)2)(x+(s2-1)2) for a rational number s ≠ 0, ± 1, and define A consisting of all pairs of a rational number s and a non-torsion rational point (α, β) ∈ E1,s(ℚ). We construct a surjective map from A to the set \mathscrF of equivalence classes of rational face cuboids, and prove that this map is a 32:1-map. In this way, we show that the set \mathscrF has infinite elements. Also, we prove that there are infinitely many s ∈ ℚ ∖ \ 0,± 1 \ with rank E1,s (ℚ)>0. In this proof, we construct pairs of s and (α, β) ∈ E1,s (ℚ) which are not parametric solutions.