Open problem, Arithmetic and number theory, posed 1993

Beal conjecture

Open

If $A^x + B^y = C^z$ where $A, B, C, x, y, z$ are positive integers with $x, y, z > 2$ , then $A, B,$ and $C$ have a common prime factor.

Research frontier as of 2026

As of 2026, the Beal conjecture remains open and the USD $1{,}000{,}000$ AMS prize is unclaimed. By combining Wiles's modularity method (Frey curves and Ribet's level-lowering theorem) with Chabauty–Coleman $p$ -adic integration and hyperelliptic curve techniques, mathematicians have proved the absence of coprime solutions for many infinite families of exponent triples $(x, y, z)$ , including $(p, p, 2)$ , $(p, p, 3)$ , and $(2, 3, n)$ for various ranges of $n$ (after accounting for the known Catalan solution $1^n + 2^3 = 3^2$ ). However, treating three independent variable exponents $(x, y, z)$ simultaneously lies beyond current Frey-curve constructions.

Best known results

Darmon and Granville (1995): for any fixed triple $(x, y, z)$ with $1/x + 1/y + 1/z < 1$ , there are only finitely many coprime integer solutions to $A^x + B^y = C^z$ .
Modular and Chabauty methods completely solve many families of signatures such as $(p, p, 2)$ and $(p, p, 3)$ for primes $p \ge 3$ , as well as $(3, 3, n)$ and $(4, 4, n)$ .

Tools and where they stop

Tool	Achieved	Where it stops
Modularity and Frey–Hellegouarch curves	Proves Fermat's Last Theorem ( $x = y = z \ge 3$ ) and rules out coprime solutions for families like $(p, p, 2)$ and $(p, p, 3)$	When all three exponents $x, y, z$ vary independently, no single Frey curve over $\mathbb{Q}$ is known that attaches to every triple
Faltings's theorem via branched coverings (Darmon–Granville)	Proves finiteness of coprime solutions for every individual fixed exponent triple $(x, y, z)$ with $1/x + 1/y + 1/z < 1$	Faltings's theorem is ineffective and applies one triple $(x, y, z)$ at a time, rather than uniformly across all exponents

Open questions

Does there exist any coprime integer solution to $A^x + B^y = C^z$ with $x, y, z \ge 3$ ?
Can Frey representations over totally real fields or hypergeometric motives handle three independent prime exponents $(p, q, r)$ ?

References

R. Daniel Mauldin (1997). A generalization of Fermat's Last Theorem: The Beal Conjecture and Prize Problem
Henri Darmon, Andrew Granville (1995). On the equations z^m = F(x, y) and Ax^p + By^q = Cz^r · DOI:10.1112/blms/27.6.513
Henri Cohen (2007). Number Theory, Volume II: Analytic and Modern Tools · DOI:10.1007/978-0-387-49894-2