Conjecture Fieldbook

GlossaryStudy cards

Perfect cuboid

Canonical problem dossier

Does a rectangular box exist whose three edge lengths, three face diagonals, and space diagonal are all positive integers?

Status
Open
Field
Diophantine equations
Verification
Reviewed dossier
Source grade
secondary
Reviewed
2026-07-27T18:04:33.834Z
Reviewer
Conjecture Fieldbook maintainers

Formal expression

a2+b2=d2, a2+c2=e2, b2+c2=f2, a2+b2+c2=g2a^2+b^2=d^2,\ a^2+c^2=e^2,\ b^2+c^2=f^2,\ a^2+b^2+c^2=g^2

Why it matters

The problem is an accessible but stubborn system of simultaneous Diophantine equations.

Context and known approaches

Euler bricks satisfy the three face-diagonal conditions; adding the space-diagonal condition creates the unresolved perfect-cuboid problem.

Formal boundary

Find positive integers a,b,c,d,e,f,g satisfying a²+b²=d², a²+c²=e², b²+c²=f², and a²+b²+c²=g², or prove none exist.

Known partial results

Pending review. No structured partial-result note has been added to this dossier.

Source trail

  1. Wikipedia — Definitions and known constraints.
  2. Wikipedia open-problems list — Composite index by mathematical field.