[#P20] Beal conjecture
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 \(\gcd(A,B,C)>1\).
1Context
The conjecture is a generalized Fermat-type problem and has strong connections to the abc conjecture.
2Problem setup
Definition 1 (A common prime factor). A common prime factor is a prime that divides each of A, B, and C.
Definition 2 (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 1. The conjecture is a generalized Fermat-type problem and has strong connections to the abc conjecture.
3What 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.
1Status
Current status (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.[1][2]
1Packet records
Recent contributions
These records are attached to this problem after the current published packet. Each badge shows its current verification or packet-review step.
Notes and companion material
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 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Consult the AMS prize procedures and the literature on generalized Fermat equations before presenting a claimed solution.
Recorded example 1. 3^3 + 6^3 = 3^5 satisfies the equation, and the three bases share the prime factor 3.
Computational notes
- A bounded search can exclude counterexamples only within its tested ranges of bases and exponents.
2See also
- Rational points on y^2=x^6-x^2+1diophantine equations
- Square-class collisions in the Pell-Lucas sequencediophantine equations
- Integral-root classification for Lloyd polynomialsdiophantine equations
How to cite
TheoremDB contributors, “Beal conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/beal-conjectureThis page as plain text: beal-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from ams.org[1], current as of July 31, 2026.
1References
- Packet source. 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. ↗website · primary source · checked 2026-07-31Source 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.Also cited at problem and prize statement.Also cited at Editorial research route recorded 2026-07-31.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.
- The Beal Conjecture, official prize and problem site, checked 2026-08-01. conjecture statement and prize information. ↗website · primary source · checked 2026-08-01Source use: original summary.Confirms that the prize still seeks a proof or disproof and restates the exact target.
An original CC0 restatement prepared by TheoremDB maintainers.