Nonexistence of Euler Boxes with Semiprime Sides

TL;DR

The paper proves that a perfect Euler box cannot have a semiprime side length by reducing the problem to elementary divisor relations of $a^2$ with $a=pq$, then performing a case analysis based on how the difference-of-squares factors $(d-b,d+b)$ and $(e-c,e+c)$ can occur. Through two main cases—symmetric and asymmetric in $p$ and $q$—the authors derive explicit expressions for $b$ and $c$ and show that all feasible divisor configurations lead to contradictions with the fundamental diagonals relations, specifically $d^2=a^2+b^2$, $e^2=a^2+c^2$, $f^2=b^2+c^2$, and $g^2=a^2+b^2+c^2$. The result strengthens known constraints on Euler boxes, and the method suggests a broader framework for ruling out further semiprime or higher-prime-edge possibilities, including a new proof that an edge cannot be prime length. The work also discusses potential generalizations to sides with more prime factors and future directions for tightening constraints on perfect Euler boxes.

Abstract

In this paper we prove that there cannot exist a perfect Euler box with a semiprime side. We first display the proof, which uses nothing more than elementary number theory. Due to the elementary nature of this proof, it is possible that more complex techniques could be used to generalize it to a stronger constraint on Euler boxes; this is discussed at the end.

Figures (1)

• Figure 1: Illustration of an Euler brick. WikipediaEN:AFM

Theorems & Definitions (8)

• Theorem 1.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 4.1