# P12771: Does Substitution by x^2+x+1 Preserve All-Ones Irreducibility Exponents?

- ID: `P12771`
- Reference: `does-substitution-by-x-2-x-1-preserve-all-ones-irreducibility-exponents`
- Page: https://theoremdb.org/statements/P12771
- 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/does-substitution-by-x-2-x-1-preserve-all-ones-irreducibility-exponents?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(a(n)\) be Numbers n such that x^n + x^(n-1) + x^(n-2) + ... + x + 1 is irreducible over GF(2). Is it true that terms >= 2 are numbers n such that P^n + P^(n-1) + P^(n-2) + ... + P + 1 is irreducible over GF(2), where P=x^2+x+1?

## Status

**Resolution status:** Open.
**Lean verification:** Not Lean-verified.

A Lean formalization is attached. Signed verification is pending.

## 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 `does-substitution-by-x-2-x-1-preserve-all-ones-irreducibility-exponents`, 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.