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Week 5 · ClassicalOpen

Beal's Conjecture — The Million-Dollar Generalization of Fermat

Research level· Posed 1993

Problem

If $A^x + B^y = C^z$ in positive integers with $x, y, z > 2$ , then $A$ , $B$ , $C$ share a common prime factor. A generalization of Fermat's Last Theorem (case $x=y=z$ ).

Why it matters

Wiles' proof of Fermat's Last Theorem was published in 1995. Andrew Beal formulated the different-exponent conjecture in 1993; a $5,000 prize was announced in 1997 and was raised to$ 1,000,000 in 2013. Frey curves and modular forms do not by themselves settle this broader problem.

Progress so far

Finite searches have excluded several large but differently shaped regions. Norvig records a distributed search with bases $A,B$ up to 200,000 and exponents up to 5,000, as well as slices such as bases up to 250,000 with exponents up to 7. These are finite negative searches, not a proof. The AMS administers the $1,000,000 prize.

💡 Explore together, one line at a time(0 contributions)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

You do not need a complete proof. A small observation can open the next path.

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