# P11580: Linear complexity of binary Sierpinski-triangle rows

- ID: `P11580`
- Reference: `linear-complexity-of-binary-sierpinski-triangle-rows`
- Page: https://theoremdb.org/statements/P11580
- 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/linear-complexity-of-binary-sierpinski-triangle-rows?representation=markdown
- Record maturity: Reviewed problem

## The problem

Let \(L(m)\) be the linear complexity over \(\operatorname{GF}(2)\) of the binary digits of \(m\), read from most significant to least significant. Thus \(L(m)\) is the degree of the shortest linear feedback shift-register recurrence satisfied by that finite word. Let \(R(n)\) be row \(n\) of Pascal's triangle reduced modulo \(2\) and read as one binary integer. Prove that for every \(n\ge 1\), \(L(R(n))=2^{\lfloor\log_2 n\rfloor+1}-n\).

## 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 `linear-complexity-of-binary-sierpinski-triangle-rows`, 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.