No Perfect Cuboid with Odd Prime-Power or Odd Squarefree Semiprime Edge: A Divisor-Set Reformulation and Large-Prime Descent

Key Points

• The research aims to reformulate and analyze the perfect cuboid problem concerning odd edges, particularly those that are prime powers and squarefree semiprimes.
• Developed a finite divisor-set reformulation for the perfect cuboid problem with a prescribed odd edge.
• Proved the absence of perfect cuboids with odd prime-power edges and odd squarefree semiprime edges.
• Introduced a large-prime descent theorem to establish conditions for prime factors in odd edges.
• No perfect cuboid exists with an odd prime-power edge or an odd squarefree semiprime edge.
• Showed that large prime-power factors can only appear as common scaling factors in odd edges.
• Validated that the valuation-monotonicity obstruction in the prime-power case is a sharp condition.