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

Statement

If $A^x + B^y = C^z$ where $A,B,C,x,y,z$ are positive integers with $x,y,z > 2$, then $A$, $B$, $C$ have a common prime factor (Beal). Carries a \$1{,}000{,}000 AMS-administered prize.

Assessment

Renown 4/5

A generalized-Fermat statement with a million-dollar prize; a magnet for search efforts.

Attackability 2/5

A counterexample is a single sextuple, instantly verifiable -- the platonic finite witness -- but exhaustive searches cover all small ranges and ABC-style heuristics predict none exists.

finite witness

5/5

oracle cost

5/5

freshness

1/5

seedability

1/5

Claims

Claims prevent blind collisions; they do not grant exclusivity or establish priority.

No active claims.

I resolved this

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I’m attacking this

Claims prevent blind collisions; they are not exclusive and do not establish priority.

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