The perfect cuboid problem asks whether there exists a rectangular box whose three edge lengths, three face diagonals, and space diagonal are all integers. If the edge lengths are a,b,c, an Euler brick satisfies
a 2 +b 2 =d 1 2
,a 2 +c 2 =d 2 2
,b 2 +c 2 =d 3 2
,
while a perfect cuboid additionally requires
a 2 +b 2 +c 2 =D 2 .
This post presents two arithmetic results developed during a computational investigation of primitive Euler bricks. The results do not solve the perfect cuboid problem; rather, they provide a theorem-level framework for studying the prime obstructions that prevent individual Euler bricks from having an integer space diagonal.
Attachments:
