# P11945: Doubling law for rotated XOR cycle lengths

- ID: `P11945`
- Reference: `doubling-law-for-rotated-xor-cycle-lengths`
- Page: https://theoremdb.org/statements/P11945
- 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/doubling-law-for-rotated-xor-cycle-lengths?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(a(n)\) be the cycle length of the \(n\)-bit process in OEIS A337742: start with an all-zero previous word and a current word having one initial \(1\); repeatedly apply XOR to the two words, rotate the result cyclically by one bit, and shift the current word into the previous slot. Prove that \(a(2n)=2a(n)\) for every \(n\ge 1\).

## Status

**Resolution status:** Open.

Research work is recorded. A complete resolution has not passed review.

## 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 `doubling-law-for-rotated-xor-cycle-lengths`, 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.