# P14: Perfect cuboid problem

- ID: `P14`
- Reference: `perfect-cuboid-problem`
- Page: https://theoremdb.org/statements/P14
- Record maturity: Reviewed problem with recorded work

## Problem

A perfect cuboid exists: there are positive integers \(a,b,c\) for which the four numbers \(\sqrt{a^2+b^2}\), \(\sqrt{a^2+c^2}\), \(\sqrt{b^2+c^2}\), and \(\sqrt{a^2+b^2+c^2}\) are integers.

### Context

Euler bricks exist, but no example is known whose space diagonal is also integral.

### Problem setup

- **Definition (A perfect cuboid).** A perfect cuboid is a rectangular box whose three edge lengths, three face diagonals, and space diagonal are positive integers.
- **Definition (An Euler brick has integer edges and face diagonals, with no requirement on the space diagonal).** An Euler brick has integer edges and face diagonals, with no requirement on the space diagonal.
- **Remark.** Euler bricks exist, but no example is known whose space diagonal is also integral.

### What counts as a solution

- Exhibit positive integers satisfying all four square conditions, or prove that no positive integer triple can satisfy them.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: Euler bricks provide positive integer edges and all three integral face diagonals. The integral space diagonal remains the missing condition. Exact unresolved remainder: Find positive integers a,b,c for which all three face diagonals and the space diagonal are integers, or prove no such triple exists. [1](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: Euler bricks provide positive integer edges and all three integral face diagonals. The integral space diagonal remains the missing condition. Exact unresolved remainder: Find positive integers a,b,c for which all three face diagonals and the space diagonal are integers, or prove no such triple exists.

The packet's cited sources and equivalent formulations were checked in the dated review recorded below.

Strongest checked result: Euler bricks provide positive integer edges and all three integral face diagonals. The integral space diagonal remains the missing condition.

Exact unresolved remainder: Find positive integers a,b,c for which all three face diagonals and the space diagonal are integers, or prove no such triple exists.

### Background and intake notes

- Original intake status: The cited scholarly paper describes the existence or nonexistence of a perfect cuboid as a famous unsolved problem. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited paper on 2026-07-22.
- Many parametrizations, congruence restrictions, and large searches are known. Claimed nonexistence proofs require especially careful review.

- Recorded example: The edges 44, 117, and 240 form an Euler brick with integer face diagonals 125, 244, and 267; its space diagonal is not an integer.

### Open directions

- **Route 1** (reported): Exhibit positive integers satisfying all four square conditions, or prove that no positive integer triple can satisfy them. [1](#reference-1)

### Computational notes

- A bounded search can exclude candidates only within its parameter range.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `perfect-cuboid-problem`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Ruslan Sharipov, “On Walter Wyss's no perfect cuboid paper”. arXiv:1704.00165 (2017). Ruslan Sharipov, arXiv:1704.00165, abstract and analysis of a claimed resolution https://arxiv.org/abs/1704.00165
   - Also cited at Abstract and error analysis
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:1704.00165, checked 2026-07-31; checked 2026-07-31
   - Source use: original_summary
   - The cited scholarly paper describes the existence or nonexistence of a perfect cuboid as a famous unsolved problem. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
   - Source used to formulate or check the problem record.
   - Source used to assess the problem's recorded status.
   - Packet-linked scholarly status source and critique of a claimed proof.
   - Source named by the research packet.