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Week 5 · ClassicalOpen

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

Research level· Posed 1997

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

Fermat's Last Theorem (case $x=y=z$ ) was solved by Wiles (1995) after 358 years. In 1997 Texas banker–amateur Andrew Beal generalized it to different exponents. The prize started at $5{,}000 in 1997 and was raised to *$ 1{,}000{,}000 in 2013*. Frey curves and modular forms (Wiles' tools) don't apply directly — new ideas are needed.

Progress so far

Verified for $A,B,C \le 1000$ and $x,y,z \le 7$ (Norvig and others). Partial results for some special exponent combinations. The prize is administered by the AMS — paid for either a proof or a counterexample.

💡 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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