Euler Perfect Box 2nd Way

This work presents a complete proof demonstrating the nonexistence of an Euler Perfect Box (EPB)—a rectangular parallelepiped in which all edges, face diagonals, and the space diagonal are positive integers. By examining three exhaustive structural cases—(A) (a < b < c), (B) (a = b < c), and (C) (a = b = c)—the analysis shows that at least one diagonal in each configuration becomes non integer. Through algebraic substitution and comparison of squared diagonal expressions, the proof establishes that no arrangement of integer edges can satisfy all required diagonal conditions simultaneously. Therefore, an Euler Perfect Box cannot exist. This result contributes to the broader study of Diophantine geometry and extends the classical investigation of Euler bricks.