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Statement

Prove that whenever

A^x + B^y = C^z

holds for positive integers

A, B, C

and integer exponents

x, y, z \geq 3

, the bases

A, B, C

share a common prime factor (i.e.,

\gcd(A, B, C) > 1

).

Current frontier

Open in general (AMS $1,000,000 prize active); resolved for specific exponent triples — Fermat-Wiles (1995) handles x = y = z, and the generalized Fermat equation has been settled for many individual signatures (e.g., (2,3,7), (2,3,10), (2,3,11), (3,4,5)) via modular methods.

When this counts as solved

A rigorous proof that no coprime solution exists for any exponent triple

(x, y, z)

with

x, y, z \geq 3

, OR an explicit verified counterexample: positive integers

A, B, C

with

\gcd(A, B, C) = 1

and

A^x + B^y = C^z

for some

x, y, z \geq 3

. The AMS \$1,000,000 Beal prize is active on this. Resolving additional individual exponent signatures via modular / Frey-curve methods, extending computational verification, or proving stronger ABC-conditional results don't close it — the conjecture is binary.

Classification

binary