# P2422: Nonvanishing of Baum-Sweet Hankel determinants

- ID: `P2422`
- Reference: `baum-sweet-hankel-nonvanishing`
- Page: https://theoremdb.org/statements/P2422
- Record maturity: Reviewed problem with recorded work

## Problem

Let \(b_n\) be the Baum-Sweet sequence, so \(b_n = 1\) when the binary expansion of \(n\) contains no block of consecutive zeros of odd length and \(b_n = 0\) otherwise, with \(b_0 = 1\). Let \(H_n = \det(b_{i+j})_{0 \le i,j < n}\). Is \(H_n \ne 0\) for every \(n \ge 1\)?

### Problem setup

- **Remark.** The Baum-Sweet sequence is 2-automatic and arises from the Baum-Sweet power series over a finite field.
- **Definition.** H_n is the order-n Hankel determinant built from the sequence.

### What counts as a solution

- Prove that H_n is nonzero for every n at least 1, or exhibit an n with H_n = 0.

## Status

Exact integer elimination certifies \(H_n\ne0\) for \(1\le n\le110\), and a self-reported audit modulo \(100000007\) finds nonzero residues through \(n=4999\); integer nonvanishing for every \(n\ge5000\) remains open. [3](#reference-3)

## Work

### Evidence for the current status

**Proposition 1 (Non-apwenian does not mean that a determinant vanishes).** Exact integer elimination certifies \(H_n\ne0\) for \(1\le n\le110\), and a self-reported audit modulo \(100000007\) finds nonzero residues through \(n=4999\); integer nonvanishing for every \(n\ge5000\) remains open.

An apwenian binary sequence has every normalized Hankel determinant odd. Guo and Han's Example 11 classifies Baum-Sweet as non-apwenian. This agrees with \(H_3=-2\), but it leaves integer nonvanishing open. The classical function-field continued fraction also concerns a different object. Neither result resolves the candidate's claim.

### Background and intake notes

Screened in the same sweep as the Rudin-Shapiro question.

- Original intake status: Status not established. Hankel determinants of automatic sequences are actively studied and this case may be known; no search was performed.
- No literature search was carried out.
- A 0-1 sequence with no vanishing Hankel determinant at all is a stronger pattern than the Rudin-Shapiro case, which has three exceptions.

- Recorded example: H_1 = b_0 = 1.

### Other known results

- **Proposition 2** (supported): Its generating series satisfies \(B(z)=B(z^4)+zB(z^2)\), and over \(\mathbb F_2\) it is the Baum-Sweet cubic. [1](#reference-1)

### Open directions

- **Route 1** (conjectured): The next search starts at order 5,000, while a proof must control even nonzero determinants as well as odd ones. [2](#reference-2)

### Runnable artifacts

- **Artifact 1** (reproduced): Exact elimination finds no vanishing Baum-Sweet determinant through \(n=110\).
- **Artifact 2** (reported): Elimination modulo \(100000007\) reports a nonzero residue at every order through 4,999.

### Computational notes

- Exact integer Hankel determinants for all n from 1 to 110, with no vanishing value.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `baum-sweet-hankel-nonvanishing`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Leonard E. Baum and Melvin M. Sweet, Continued Fractions of Algebraic Power Series in Characteristic 2, Annals of Mathematics 103 (1976), pages 593-610. Leonard E. Baum and Melvin M. Sweet, Continued Fractions of Algebraic Power Series in Characteristic 2, Annals of Mathematics 103 (1976), pages 593-610 https://annals.math.princeton.edu/1976/103-3/p12
   - journal_article; primary source; web version checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Defines the Baum–Sweet sequence and gives the algebraic generating-function structure used in the determinant problem.
   - Source named by the research packet.
2. <a id="reference-2"></a>W.H Mills and David P Robbins, “Continued fractions for certain algebraic power series”. Journal of Number Theory 23(3) (1986), 388-404. DOI 10.1016/0022-314X(86)90083-1. Mills and Robbins, Continued fractions for certain algebraic power series, Journal of Number Theory 23 (1986), pages 388-404 https://doi.org/10.1016/0022-314X(86)90083-1
   - journal_article; primary source; version of record; checked 2026-08-01
   - Source use: original_summary
   - Develops continued fractions for the algebraic power series neighboring the Baum–Sweet Hankel problem.
3. <a id="reference-3"></a>Yining Guo and Guo-Niu Han, On a family of automatic apwenian sequences, Discrete Mathematics 348 (2025), Example 11. Yining Guo and Guo-Niu Han, On a family of automatic apwenian sequences, Discrete Mathematics 348 (2025), Example 11 https://irma.math.unistra.fr/~guoniu/papers/p120autoapw.pdf
   - journal_article; primary source; PDF checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Classifies Baum–Sweet as non-apwenian, a parity result that does not decide integer determinant nonvanishing.