# P12797: Bounded prime-factor complexity in exponential ternary equations

- ID: `P12797`
- Reference: `bounded-omega-exponential-ternary-equations`
- Page: https://theoremdb.org/statements/P12797
- Export scope: built Markdown snapshot. The current public packet may have changed since this build.
- Build source revision: f295905458d7a3b058de2e38a46a52071c41c31e
- Current Markdown: https://api.theoremdb.org/v1/statements/bounded-omega-exponential-ternary-equations?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(\Omega(n)\) denote the total number of prime divisors of \(n\), counted with multiplicity. Determine whether there exists an absolute constant \(C>0\) such that every positive integer solution \((a,b,c,x,y,z)\) of \(a^x+b^y=c^z\), with \(x,y,z>2\), satisfying \(\Omega(abc)/(x+y+z)\le C\), necessarily satisfies \(\gcd(a,b,c)=\min(a,b,c)\).

## Status

**Resolution status:** Open.

No proof or formalization has been recorded.

## Research packet

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `bounded-omega-exponential-ternary-equations`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.