[#P2420] Nonvanishing of Rudin-Shapiro Hankel determinants

Problem. Let $r_n = (-1)^{c(n)}$ where $c(n)$ counts occurrences of the block 11 in the binary expansion of $n$, and let $H_n = \det(r_{i+j})_{0 \le i,j < n}$. Is $H_n \ne 0$ for every $n \ge 9$?

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Definitions and notation

1Context

Screened alongside the Thue-Morse, Baum-Sweet, Stern, period-doubling and Cantor sequences. The Thue-Morse case is a known theorem and was discarded for that reason.

2Problem setup

Remark 1. The Rudin-Shapiro sequence takes values +1 and -1 and is generated by counting overlapping occurrences of 11 in the binary expansion.

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 all n at least 9, or exhibit some n at least 9 with H_n = 0.

1Status

What counts as a solution

Current status (Two residue classes are settled 2-adically). The published 2-adic formula proves $H_n\ne0$ for $n\equiv0,1\pmod3$, while the remaining universal case $H_{3m+2}\ne0$ for every $m\ge3$ remains open despite reported modular certificates through order 5,000.^[2]

1Packet records

Show recorded summaries

Record	Kind	Assessment
The published 2-adic formula proves $H_n\ne0$ for $n\equiv0,1\pmod3$, while the remaining universal case $H_{3m+2}\ne0$ for every $m\ge3$ remains open despite reported modular certificates through order 5,000.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 The published 2-adic formula proves $H_n\ne0$ for $n\equiv0,1\pmod3$, while the remaining universal case $H_{3m+2}\ne0$ for every $m\ge3$ remains open despite reported modular certificates through order 5,000.^[2] Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-claim-two-residue-classes (“Two residue classes are settled 2-adically”) records a bound, answer, status fact, or structural consequence. The record states: The published 2-adic formula proves $H_n\ne0$ for $n\equiv0,1\pmod3$, while the remaining universal case $H_{3m+2}\ne0$ for every $m\ge3$ remains open despite reported modular certificates through order 5,000. Evidence SupportedBacked by a cited source or by evidence short of a proof. Scope all positive orders congruent to 0 or 1 modulo 3 Sources Reference 2 Argument Put $u_n=(1-r_n)/2$ and $v_k=u_k+u_{k+2}\pmod2$. Elementary row and column operations give \[ \frac{H_n(r)}{(-2)^{n-1}}\equiv H_{n-1}(v)\pmod2. \] The generating series $V(x)=\sum v_kx^k$ obeys \[ x^2(1+x)V^2+(1+x)^2V+x=0 \] over $\mathbb F_2$. Its periodic Hankel continued fraction has valuation parameters $(1,0)^*$. Han's Theorem 2.1 then says that $H_m(v)$ is nonzero exactly when $m\not\equiv1\pmod3$. Thus the displayed quotient is odd for $n\equiv0,1\pmod3$. This reduction is an editorial derivation from Han's theorem and should receive independent proof review. It reduces the original question to orders $n=3m+2$, beginning with $n=11$. Open full record →
Han's periodic formula concerns the 0/1 Rudin-Shapiro sequence, while the candidate uses signs in $\{-1,1\}$.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 2 Han's periodic formula concerns the 0/1 Rudin-Shapiro sequence, while the candidate uses signs in $\{-1,1\}$.^[1] Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-claim-signed-binary-distinction (“Published binary formulas do not settle the signed determinant”) records a bound, answer, status fact, or structural consequence. The record states: Han's periodic formula concerns the 0/1 Rudin-Shapiro sequence, while the candidate uses signs in $\{-1,1\}$. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope the literature comparison for the signed sequence in the candidate Sources Reference 1 Argument Han's Proposition 1.3 computes Hankel determinants modulo 2 for the binary sequence $u_n=(1-r_n)/2$. Adamczewski and Rivoal construct selected Padé approximants for the signed generating function. Neither cited result states eventual nonvanishing of the leading signed Hankel minors. The signed and binary questions need separate records because reducing entries modulo 2 turns the signed Hankel matrix into an all-ones matrix. Related: Proposition 1 Open full record →
The open core is $H_{3m+2}\neq0$ for every $m\geq3$.	Trace	Evidence: Conjectured Proposed as likely true. No argument is recorded.
attempt · Route 1 The open core is $H_{3m+2}\neq0$ for every $m\geq3$.^[3] Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-attempt-residue-class-two (“Settle the remaining orders congruent to 2 modulo 3”) documents a concrete method, search boundary, or failed route. The record states: The open core is $H_{3m+2}\neq0$ for every $m\geq3$. Evidence ConjecturedProposed as likely true. No argument is recorded. Record state open strategy Scope No narrower scope recorded Sources Reference 3 What happened The 2-adic argument becomes inconclusive in this residue class because the reduced determinant is even. A useful next step is a continued-fraction or modular recurrence that treats $H_{3m+2}$ directly. Any counterexample search should begin above the certified bound 5,000. Related: Proposition 1 · Artifact 2 Open full record →
Bareiss elimination finds zeros exactly at orders $2,5,8$ through $n=110$.	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Bareiss elimination finds zeros exactly at orders $2,5,8$ through $n=110$. Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-artifact-determinants-110 (“Exact signed determinant sweep through order 110”) supplies evidence or a replay used to check the packet. The record states: Bareiss elimination finds zeros exactly at orders $2,5,8$ through $n=110$. Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope every determinant order from 1 through 110 File research/candidate-screening/verify.py Details The checker generates $r_n=(-1)^{\operatorname{popcount}(n\mathbin{\&}(n\!\gg\!1))}$ and computes every leading Hankel determinant over the integers. The first twelve values are $1,0,-4,-8,0,32,-192,0,6912,10752,-112640,-780288$. No zero occurs after order 8 in the checked range. Related: Proposition 1 Open full record →
Five prime fields certify every order through 5,000 except the known zeros $2,5,8$.	Artifact	Evidence: Reported Stated by one agent or source, not independently checked.
artifact · Artifact 2 Five prime fields certify every order through 5,000 except the known zeros $2,5,8$. Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-artifact-modular-certificates-5000 (“Modular nonvanishing certificates through order 5,000”) supplies evidence or a replay used to check the packet. The record states: Five prime fields certify every order through 5,000 except the known zeros $2,5,8$. Evidence ReportedStated by one agent or source, not independently checked. Scope every determinant order from 1 through 5000 Details Berlekamp-Massey computations over $\mathbb F_p$, for $p=3,5,7,11,13$, certify nonzero determinants through order 5,000. The first witnessing-prime counts were 3: 3,290 orders; 5: 1,357; 7: 306; 11: 39; 13: 5. The only orders with no prime witness were 2, 5, and 8, whose integer determinants vanish. This computation was performed during the status audit; a standalone repository script has yet to be retained. Related: Artifact 1 Open full record →
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Notes and companion material

Original intake status. UNKNOWN as of 2026-07-28. A published 2-adic formula proves nonvanishing for n congruent to 0 or 1 modulo 3. The universal n≡2 modulo 3 case remains open after the checked sources and computation through n=5000.

The 2026-07-28 audit reduced the open part to the residue class n≡2 mod 3 using a published 2-adic formula.
Exact and modular packet computations find no further zero through n=5000; this remains bounded evidence.
No duplicate Rudin–Shapiro Hankel target was found in the controlled corpus.

Recorded example 1. H_2 = 0, H_5 = 0 and H_8 = 0 are the only vanishing determinants found.

Computational notes

Exact integer Hankel determinants for all n from 1 to 110. The determinant vanished exactly at n = 2, 5 and 8 and was nonzero for every other n in that range.

How the 5 records connect

The overview places each record once. The relation list includes shared dependencies and names both ends of each link.

ProblemNonvanishing of Rudin-Shapiro Hankel determinants

All 5 recorded relations between these records and the problem

2See also

Nonvanishing of Baum-Sweet Hankel determinantsautomatic sequences
Algebraicity of a Thue-Morse infinite productautomatic sequences
Ternary representatives of finite-order integral matriceslinear algebra

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Plain text

“Nonvanishing of Rudin-Shapiro Hankel determinants.” TheoremDB. P2420. Problem statement; statement text SHA-256 e7149c799fc881382cb8347fca62780ea56b8faa5aad848b1c201c1d99e15df2. https://theoremdb.org/statement/?ref=P2420

BibTeX

@misc{theoremdb-problem-e7149c799fc881382cb8347fca62780ea56b8faa5aad848b1c201c1d99e15df2,
  title = {{Nonvanishing of Rudin-Shapiro Hankel determinants}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 e7149c799fc881382cb8347fca62780ea56b8faa5aad848b1c201c1d99e15df2},
  url = {https://theoremdb.org/statement/?ref=P2420}
}

Plain text: Built Markdown snapshot

This problem includes 5 records joined by 5 typed links, sourced from doi.org^[1], current as of July 24, 2026.

1Lean verification

1References

Packet source. Guo-Niu Han, “Hankel continued fraction and its applications”. Advances in Mathematics 303 (2016), 295-321. DOI 10.1016/j.aim.2016.08.013. Proposition 1.3, Theorem 2.1, and Algorithm 3.3. ↗scholarly publication · reference source · version of record · checked 2026-08-01Source use: citation only.Publishes the Hankel continued-fraction formula used to reduce Rudin–Shapiro nonvanishing to one residue class.Also cited at Han 2016, Proposition 1.3, pages 2-3; Adamczewski-Rivoal 2009, section 2.2 and Proposition 2.2.For Nonvanishing of Rudin-Shapiro Hankel determinants: Provides the 2-adic determinant formula and continued-fraction machinery that settle two residue classes.Source named by the research packet.Provides the 2-adic determinant formula and continued-fraction machinery that settle two residue classes.
Guo-Niu Han, Hankel continued fraction and its applications, Theorem 2.1 and Algorithm 3.3; reduction derived for this entry. irma.math.unistra.fr checked 2026-08-01. Theorem 2.1 and Algorithm 3.3. ↗website · reference source · PDF checked 2026-08-01 · checked 2026-07-24Source use: citation only.Gives the 2-adic determinant formula that settles two residue classes in the Rudin–Shapiro problem.Also cited at Guo-Niu Han, Hankel continued fraction and its applications, Theorem 2.1 and Algorithm 3.3; reduction derived for this entry.For Nonvanishing of Rudin-Shapiro Hankel determinants: The published 2-adic formula proves $H_n\ne0$ for $n\equiv0,1\pmod3$, while the remaining universal case $H_{3m+2}\ne0$ for every $m\ge3$ remains open despite reported modular certificates through order 5,000.Primary full-text source used to derive and check the residue-class reduction in the packet.
László Mérai and Arne Winterhof, “On the $N$th linear complexity of automatic sequences”. arXiv:1711.10764 (2017). Theorem 2. ↗preprint · reference source · arXiv:1711.10764v1 · checked 2026-07-24Source use: citation only.Gives a neighboring automatic-sequence complexity consequence of nonzero Rudin–Shapiro Hankel determinants.Also cited at Mérai and Winterhof, On the Nth linear complexity of automatic sequences, Theorem 2.For Nonvanishing of Rudin-Shapiro Hankel determinants: The open core is $H_{3m+2}\neq0$ for every $m\geq3$.Gives a neighboring complexity consequence of nonzero Hankel determinants for the Rudin–Shapiro sequence.

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

[#P2420] Nonvanishing of Rudin-Shapiro Hankel determinants

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3What counts as a solution

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Record	Kind	Assessment
The published 2-adic formula proves \(H_n\ne0\) for \(n\equiv0,1\pmod3\), while the remaining universal case \(H_{3m+2}\ne0\) for every \(m\ge3\) remains open despite reported modular certificates through order 5,000.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 1 The published 2-adic formula proves \(H_n\ne0\) for \(n\equiv0,1\pmod3\), while the remaining universal case \(H_{3m+2}\ne0\) for every \(m\ge3\) remains open despite reported modular certificates through order 5,000.^[2] Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-claim-two-residue-classes (“Two residue classes are settled 2-adically”) records a bound, answer, status fact, or structural consequence. The record states: The published 2-adic formula proves \(H_n\ne0\) for \(n\equiv0,1\pmod3\), while the remaining universal case \(H_{3m+2}\ne0\) for every \(m\ge3\) remains open despite reported modular certificates through order 5,000. Evidence SupportedBacked by a cited source or by evidence short of a proof. Scope all positive orders congruent to 0 or 1 modulo 3 Sources Reference 2 Argument Put \(u_n=(1-r_n)/2\) and \(v_k=u_k+u_{k+2}\pmod2\). Elementary row and column operations give \[ \frac{H_n(r)}{(-2)^{n-1}}\equiv H_{n-1}(v)\pmod2. \] The generating series \(V(x)=\sum v_kx^k\) obeys \[ x^2(1+x)V^2+(1+x)^2V+x=0 \] over \(\mathbb F_2\). Its periodic Hankel continued fraction has valuation parameters \((1,0)^*\). Han's Theorem 2.1 then says that \(H_m(v)\) is nonzero exactly when \(m\not\equiv1\pmod3\). Thus the displayed quotient is odd for \(n\equiv0,1\pmod3\). This reduction is an editorial derivation from Han's theorem and should receive independent proof review. It reduces the original question to orders \(n=3m+2\), beginning with \(n=11\). Open full record →
Han's periodic formula concerns the 0/1 Rudin-Shapiro sequence, while the candidate uses signs in \(\{-1,1\}\).	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Proposition 2 Han's periodic formula concerns the 0/1 Rudin-Shapiro sequence, while the candidate uses signs in \(\{-1,1\}\).^[1] Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-claim-signed-binary-distinction (“Published binary formulas do not settle the signed determinant”) records a bound, answer, status fact, or structural consequence. The record states: Han's periodic formula concerns the 0/1 Rudin-Shapiro sequence, while the candidate uses signs in \(\{-1,1\}\). Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state established Scope the literature comparison for the signed sequence in the candidate Sources Reference 1 Argument Han's Proposition 1.3 computes Hankel determinants modulo 2 for the binary sequence \(u_n=(1-r_n)/2\). Adamczewski and Rivoal construct selected Padé approximants for the signed generating function. Neither cited result states eventual nonvanishing of the leading signed Hankel minors. The signed and binary questions need separate records because reducing entries modulo 2 turns the signed Hankel matrix into an all-ones matrix. Related: Proposition 1 Open full record →
The open core is \(H_{3m+2}\neq0\) for every \(m\geq3\).	Trace	Evidence: Conjectured Proposed as likely true. No argument is recorded.
attempt · Route 1 The open core is \(H_{3m+2}\neq0\) for every \(m\geq3\).^[3] Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-attempt-residue-class-two (“Settle the remaining orders congruent to 2 modulo 3”) documents a concrete method, search boundary, or failed route. The record states: The open core is \(H_{3m+2}\neq0\) for every \(m\geq3\). Evidence ConjecturedProposed as likely true. No argument is recorded. Record state open strategy Scope No narrower scope recorded Sources Reference 3 What happened The 2-adic argument becomes inconclusive in this residue class because the reduced determinant is even. A useful next step is a continued-fraction or modular recurrence that treats \(H_{3m+2}\) directly. Any counterexample search should begin above the certified bound 5,000. Related: Proposition 1 · Artifact 2 Open full record →
Bareiss elimination finds zeros exactly at orders \(2,5,8\) through \(n=110\).	Artifact	Evidence: Executable material Executable source is recorded. Inspect its inputs and instructions before attempting a replay.
artifact · Artifact 1 Bareiss elimination finds zeros exactly at orders \(2,5,8\) through \(n=110\). Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-artifact-determinants-110 (“Exact signed determinant sweep through order 110”) supplies evidence or a replay used to check the packet. The record states: Bareiss elimination finds zeros exactly at orders \(2,5,8\) through \(n=110\). Evidence Executable materialExecutable source is recorded. Inspect its inputs and instructions before attempting a replay. Record state available Scope every determinant order from 1 through 110 File research/candidate-screening/verify.py Details The checker generates \(r_n=(-1)^{\operatorname{popcount}(n\mathbin{\&}(n\!\gg\!1))}\) and computes every leading Hankel determinant over the integers. The first twelve values are \(1,0,-4,-8,0,32,-192,0,6912,10752,-112640,-780288\). No zero occurs after order 8 in the checked range. Related: Proposition 1 Open full record →
Five prime fields certify every order through 5,000 except the known zeros \(2,5,8\).	Artifact	Evidence: Reported Stated by one agent or source, not independently checked.
artifact · Artifact 2 Five prime fields certify every order through 5,000 except the known zeros \(2,5,8\). Relevance to this problem For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-artifact-modular-certificates-5000 (“Modular nonvanishing certificates through order 5,000”) supplies evidence or a replay used to check the packet. The record states: Five prime fields certify every order through 5,000 except the known zeros \(2,5,8\). Evidence ReportedStated by one agent or source, not independently checked. Scope every determinant order from 1 through 5000 Details Berlekamp-Massey computations over \(\mathbb F_p\), for \(p=3,5,7,11,13\), certify nonzero determinants through order 5,000. The first witnessing-prime counts were 3: 3,290 orders; 5: 1,357; 7: 306; 11: 39; 13: 5. The only orders with no prime witness were 2, 5, and 8, whose integer determinants vanish. This computation was performed during the status audit; a standalone repository script has yet to be retained. Related: Artifact 1 Open full record →
No records match these filters.

1Context

2Problem setup

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance