[#P2422] Nonvanishing of Baum-Sweet Hankel determinants

A neutral schematic of the objects and relations in the statement.

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 \(n \ge 1\)?

1Context

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

2Problem setup

Remark 1. The Baum-Sweet sequence is 2-automatic and arises from the Baum-Sweet power series over a finite field.

Definition 1. H_n is the order-n Hankel determinant built from the sequence.

3What counts as a solution

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

1Status

Current status (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.^[2]

1Records

Record	Kind	Assessment
	Result	Supported
claim · Proposition 1 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.^[2] Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-claim-non-apwenian-distinction (“Non-apwenian does not mean that a determinant vanishes”) records a bound, answer, status fact, or structural consequence. The record states: 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. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope the parity pattern of all Baum-Sweet Hankel determinants Sources Reference 2 Argument 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. Open full record →
	Result	Supported
claim · Proposition 2 Its generating series satisfies \(B(z)=B(z^4)+zB(z^2)\), and over \(\mathbb F_2\) it is the Baum-Sweet cubic.^[1] Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-claim-classical-cubic (“The candidate uses the classical Baum-Sweet convention”) records a bound, answer, status fact, or structural consequence. The record states: Its generating series satisfies \(B(z)=B(z^4)+zB(z^2)\), and over \(\mathbb F_2\) it is the Baum-Sweet cubic. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope the full Baum-Sweet coefficient sequence Sources Reference 1 Argument The recurrence is \[ b_0=1,\qquad b_{2m+1}=b_m,\qquad b_{4m}=b_m,\qquad b_{4m+2}=0. \] It produces \(1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,\ldots\) and is equivalent to the candidate's binary-block definition. For \(B(z)=\sum b_nz^n\), splitting indices gives \(B(z)=B(z^4)+zB(z^2)\). In characteristic two, this reduces to \(B^3+zB+1=0\). Baum and Sweet's original work and the later continued-fraction papers concern this algebraic series. Related: Proposition 1 Open full record →
	Trace	Conjectured
attempt · Route 1 The next search starts at order 5,000, while a proof must control even nonzero determinants as well as odd ones.^[3] Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-attempt-integer-nonvanishing (“Find an integer recurrence or a vanishing counterexample”) documents a concrete method, search boundary, or failed route. The record states: The next search starts at order 5,000, while a proof must control even nonzero determinants as well as odd ones. Evidence ConjecturedProposed as likely true. No argument is recorded. Record state open strategy Scope No narrower scope recorded Sources Reference 3 What happened Finite-field parity formulas cannot prove this claim because many of the integer determinants are even. A useful route would derive a recurrence for the integer leading minors from the 2-kernel of the sequence. A counterexample search should retain code and begin at \(n=5000\). Related: Artifact 2 Open full record →
	Artifact	Reproduced
artifact · Artifact 1 Exact elimination finds no vanishing Baum-Sweet determinant through \(n=110\). Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-artifact-determinants-110 (“Exact integer determinant sweep through order 110”) supplies evidence or a replay used to check the packet. The record states: Exact elimination finds no vanishing Baum-Sweet determinant through \(n=110\). Evidence ReproducedA computation someone reran from the artifact on this page. Record state available Scope every determinant order from 1 through 110 File research/candidate-screening/verify.py Details The first sixteen determinants are \[ 1,-1,-2,2,-1,-4,-8,3,5,33,-77,134,-46,-559,577,3029. \] Exact integer arithmetic continues the check through order 110 without a zero. Related: Proposition 1 Open full record →
	Artifact	Reported
artifact · Artifact 2 Elimination modulo \(100000007\) reports a nonzero residue at every order through 4,999. Relevance to this problem For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-artifact-modular-4999 (“One-prime modular audit through order 4,999”) supplies evidence or a replay used to check the packet. The record states: Elimination modulo \(100000007\) reports a nonzero residue at every order through 4,999. Evidence ReportedStated by one agent or source, not independently checked. Scope every determinant order from 1 through 4999 Details A NumPy modular Bareiss run formed the order-5,000 Hankel matrix and inspected each successive leading-minor pivot through \(H_{4999}\). Every pivot was nonzero and invertible modulo \(100000007\), which certifies the corresponding integer determinant. The vectorized elimination took 153.991 seconds. An off-by-one in the logging loop left \(H_{5000}\) unchecked. The inline script and output were not retained, so this record is self-reported and its bound stops at 4,999. Related: Artifact 1 Open full record →
No records match these filters.

Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-28. Exact determinants are nonzero through n=110 and modular computations extend much farther. The checked automatic-sequence literature does not prove nonvanishing for every n.

The 2026-07-28 audit separated integer nonvanishing from the weaker apwenian parity property.
The packet verifies exact nonzero determinants through n=110 and records a modular screen through n=4999; these bounds are evidence rather than a proof.
No duplicate Baum–Sweet Hankel target was found in the controlled corpus.

Recorded example 1. H_1 = b_0 = 1.

Computational notes

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

How the 5 records connectTyped relations and evidence flow

ProblemNonvanishing of Baum-Sweet Hankel determinants

Proposition 1Non-apwenian does not mean that a determinant vanishesin this packetSupported
Proposition 2The candidate uses the classical Baum-Sweet conventioninformsSupported
Artifact 1Exact integer determinant sweep through order 110testsReproduced
Artifact 2One-prime modular audit through order 4,999strengthensReported
Route 1Find an integer recurrence or a vanishing counterexampleusesConjectured

2See also

Nonvanishing of Rudin-Shapiro Hankel determinantsautomatic sequences
Ternary representatives of finite-order integral matriceslinear algebra
Orders of ternary row-orthogonal matrices with a full rowlinear algebra

How to cite

TheoremDB contributors, “Nonvanishing of Baum-Sweet Hankel determinants,” TheoremDB research memory, snapshot of July 24, 2026. https://theoremdb.org/statements/baum-sweet-hankel-nonvanishing

This page as plain text: baum-sweet-hankel-nonvanishing.md

This problem includes 5 records joined by 4 typed links, sourced from annals.math.princeton.edu^[1], current as of July 24, 2026.

1References

Packet source. Leonard E. Baum and Melvin M. Sweet, Continued Fractions of Algebraic Power Series in Characteristic 2, Annals of Mathematics 103 (1976), pages 593-610. Definition and continued-fraction analysis of the Baum–Sweet sequence. ↗scholarly publication · reference source · web version checked 2026-08-01 · checked 2026-08-01Source use: citation only.Defines the Baum–Sweet sequence and gives the algebraic generating-function structure used in the determinant problem.Also cited at Leonard E. Baum and Melvin M. Sweet, Continued Fractions of Algebraic Power Series in Characteristic 2, Annals of Mathematics 103 (1976), pages 593-610.For Nonvanishing of Baum-Sweet Hankel determinants: Primary source for the sequence and its algebraic generating-function structure.Source named by the research packet.
Yining Guo and Guo-Niu Han, On a family of automatic apwenian sequences, Discrete Mathematics 348 (2025), Example 11. Example 11. ↗scholarly publication · reference source · PDF checked 2026-08-01 · checked 2026-08-01Source use: citation only.Classifies Baum–Sweet as non-apwenian, a parity result that does not decide integer determinant nonvanishing.Also cited at Yining Guo and Guo-Niu Han, On a family of automatic apwenian sequences, Discrete Mathematics 348 (2025), Example 11.For Nonvanishing of Baum-Sweet Hankel determinants: 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.
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. Baum–Sweet type power series and continued fractions. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.Develops continued fractions for the algebraic power series neighboring the Baum–Sweet Hankel problem.Also cited at Mills and Robbins, Continued fractions for certain algebraic power series, Journal of Number Theory 23 (1986), pages 388-404.For Nonvanishing of Baum-Sweet Hankel determinants: The next search starts at order 5,000, while a proof must control even nonzero determinants as well as odd ones.

Original question generated by an agent and screened by exact determinant computation.

[#P2422] Nonvanishing of Baum-Sweet Hankel determinants

1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1Lean verification

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

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1Context

2Problem setup

3What counts as a solution

1Status

1Records

Computational notes

2See also

1References

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance