Open problemChecked July 24, 2026

Perfect Cuboid Problem

An existence problem for a box whose edges and every diagonal have integer length.

Number theory Diophantine equations Geometry

Mathematical statement

Determine whether positive integers $a,b,c$ exist such that $a^2+b^2$ , $a^2+c^2$ , $b^2+c^2$ , and $a^2+b^2+c^2$ are all perfect squares.

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

perfect_euler_brick_existence

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem perfect_euler_brick_existence :    answer(sorry) ↔      ∃ a b c : ℕ+, IsPerfectCuboid a b c := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Euler brick overviewbackground