# P20: Beal conjecture

- ID: `P20`
- Reference: `beal-conjecture`
- Page: https://theoremdb.org/statements/P20
- Record maturity: Reviewed problem with recorded work

## Problem

Whenever \(A^x+B^y=C^z\) for positive integers \(A,B,C\) and exponents \(x,y,z>2\), the bases have a common prime divisor, so \(\gcd(A,B,C)>1\).

### Context

The conjecture is a generalized Fermat-type problem and has strong connections to the abc conjecture.

### Problem setup

- **Definition (A common prime factor).** A common prime factor is a prime that divides each of A, B, and C.
- **Definition (Equivalently, the equation has no solution under the stated exponent conditions when the greatest common divisor of A, B, and C).** Equivalently, the equation has no solution under the stated exponent conditions when the greatest common divisor of A, B, and C is 1.
- **Remark.** The conjecture is a generalized Fermat-type problem and has strong connections to the abc conjecture.

### What counts as a solution

- Prove that every solution under the stated exponent conditions has a common prime factor, or give a fully verified solution whose three bases have no common prime factor.

## Status

Unresolved in this packet after the dated source check. Strongest checked result: The official prize statement continues to solicit a proof or counterexample. Exact-title and Crossref searches returned claimed proofs, but no checked official or established research source confirmed a resolution. Exact unresolved remainder: Prove that every solution with exponents greater than two has a common prime factor in the three bases, or give and verify a coprime-base counterexample. [1](#reference-1) [2](#reference-2)

## 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: The official prize statement continues to solicit a proof or counterexample. Exact-title and Crossref searches returned claimed proofs, but no checked official or established research source confirmed a resolution. Exact unresolved remainder: Prove that every solution with exponents greater than two has a common prime factor in the three bases, or give and verify a coprime-base counterexample.

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

Strongest checked result: The official prize statement continues to solicit a proof or counterexample. Exact-title and Crossref searches returned claimed proofs, but no checked official or established research source confirmed a resolution.

Exact unresolved remainder: Prove that every solution with exponents greater than two has a common prime factor in the three bases, or give and verify a coprime-base counterexample.

### Background and intake notes

- Original intake status: The AMS continues to administer a prize for a proof or counterexample to this conjecture. The official statement and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation was checked against the cited AMS article on 2026-07-22.
- Consult the AMS prize procedures and the literature on generalized Fermat equations before presenting a claimed solution.

- Recorded example: 3^3 + 6^3 = 3^5 satisfies the equation, and the three bases share the prime factor 3.

### Open directions

- **Route 1** (reported): Prove that every solution under the stated exponent conditions has a common prime factor, or give a fully verified solution whose three bases have no common prime factor. [1](#reference-1)

### Computational notes

- A bounded search can exclude counterexamples only within its tested ranges of bases and exponents.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `beal-conjecture`, 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>R. Daniel Mauldin, A Generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem, Notices of the AMS 44 (1997), 1436-1437. R. Daniel Mauldin, Notices of the American Mathematical Society 44 (1997), 1436-1437 https://www.ams.org/notices/199711/beal.pdf
   - Also cited at problem and prize statement
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source use: original_summary
   - The AMS continues to administer a prize for a proof or counterexample to this conjecture. The official statement 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.
   - Supplies the formal conjecture and its proof-or-counterexample acceptance boundary.
   - Source named by the research packet.
2. <a id="reference-2"></a>The Beal Conjecture, official prize and problem site, checked 2026-08-01. conjecture statement and prize information https://www.bealconjecture.com/
   - website; primary source; checked 2026-08-01
   - Source use: original_summary
   - Confirms that the prize still seeks a proof or disproof and restates the exact target.