Articles · Mathematics
What Is the Beal Conjecture?
A $1,000,000 prize for a one-line generalization of Fermat's Last Theorem.
The statement
If Ax + By = Cz, where A, B, C, x, y, and z are positive integers and x, y, z are all greater than 2, then A, B, and C must share a common prime factor.
Turned around: if A, B, and C are pairwise coprime (no two of them share a prime factor), the conjecture says there’s no solution at all once every exponent is above 2. It’s tracked in the Registry as LR-BEAL.
Where it comes from
Texas banker and mathematician Andrew Beal formulated the conjecture in 1993 while investigating generalizations of Fermat’s Last Theorem — the famous statement that An + Bn = Cn has no positive integer solutions for n > 2, proved by Andrew Wiles in 1995. Beal’s question asks what happens once you let the three exponents differ. Fermat’s theorem is the special case where x = y = z; Beal’s conjecture is the much broader claim covering every combination of exponents above 2.
The prize
Beal has personally funded a prize for a proof or a disproof, administered by the American Mathematical Society. Offered since 1997 and raised in stages, it now stands at $1,000,000, held in trust by the AMS rather than paid directly by Beal — a structure meant to guarantee the money is there regardless of what happens to the sponsor. It remains unclaimed.
Why it’s still open
Fermat’s Last Theorem took over 350 years and a genuinely new branch of mathematics (the modularity theorem for elliptic curves) to prove for its single, fixed exponent pattern. Beal’s conjecture asks for a proof covering every combination of exponents at once — an infinite family of Fermat-like statements bundled into one claim. Partial results exist for specific small exponent combinations, but no general proof, and no counterexample, has been found despite extensive computational search.
Why it matters here
The Beal conjecture is exactly the kind of record this Registry exists to track precisely: a single, formally stated open question, with a real, currently unclaimed monetary incentive behind it. The record links directly to the AMS’s own prize page, so the terms are never secondhand.