# P2420: Nonvanishing of Rudin-Shapiro Hankel determinants

- ID: `P2420`
- Reference: `rudin-shapiro-hankel-nonvanishing`
- Page: https://theoremdb.org/statements/P2420
- Record maturity: Reviewed problem with recorded work

## 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 \(n \ge 9\)?

### Problem setup

- **Remark.** The Rudin-Shapiro sequence takes values +1 and -1 and is generated by counting overlapping occurrences of 11 in the binary expansion.
- **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 all n at least 9, or exhibit some n at least 9 with H_n = 0.

## Status

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. [3](#reference-3)

## Work

### Evidence for the current status

**Proposition 1 (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.

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 \(n=11\).

### Background and intake notes

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.

- Original intake status: Status not established, and the surrounding area is actively studied, so a literature check is the first thing a solver should do. The analogous nonvanishing statement for the Thue-Morse sequence is a known theorem of Allouche, Peyriere, Wen and Wen, which is precisely why the Rudin-Shapiro case may also be known.
- No literature search was carried out.
- The exceptional set {2, 5, 8} is the interesting feature: the determinant vanishes three times and then appears never to vanish again, which is a stronger and more specific claim than mere eventual nonvanishing.

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

### Other known results

- **Proposition 2** (supported): Han's periodic formula concerns the 0/1 Rudin-Shapiro sequence, while the candidate uses signs in \(\{-1,1\}\). [2](#reference-2)

### Open directions

- **Route 1** (conjectured): The open core is \(H_{3m+2}\neq0\) for every \(m\geq3\). [1](#reference-1)

### Runnable artifacts

- **Artifact 1** (reproduced): Bareiss elimination finds zeros exactly at orders \(2,5,8\) through \(n=110\).
- **Artifact 2** (reported): Five prime fields certify every order through 5,000 except the known zeros \(2,5,8\).

### 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.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `rudin-shapiro-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>László Mérai and Arne Winterhof, “On the $N$th linear complexity of automatic sequences”. arXiv:1711.10764 (2017). Mérai and Winterhof, On the Nth linear complexity of automatic sequences, Theorem 2 https://arxiv.org/abs/1711.10764
   - preprint; primary source; arXiv:1711.10764v1; checked 2026-08-01
   - Source use: original_summary
   - Gives a neighboring automatic-sequence complexity consequence of nonzero Rudin–Shapiro Hankel determinants.
2. <a id="reference-2"></a>Guo-Niu Han, “Hankel continued fraction and its applications”. Advances in Mathematics 303 (2016), 295-321. DOI 10.1016/j.aim.2016.08.013. Han 2016, Proposition 1.3, pages 2-3; Adamczewski-Rivoal 2009, section 2.2 and Proposition 2.2 https://doi.org/10.1016/j.aim.2016.08.013
   - journal_article; primary source; version of record; checked 2026-08-01
   - Source use: original_summary
   - Publishes the Hankel continued-fraction formula used to reduce Rudin–Shapiro nonvanishing to one residue class.
   - Source named by the research packet.
3. <a id="reference-3"></a>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. Guo-Niu Han, Hankel continued fraction and its applications, Theorem 2.1 and Algorithm 3.3; reduction derived for this entry https://irma.math.unistra.fr/~guoniu/papers/p94hfrac.pdf
   - website; primary source; PDF checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Gives the 2-adic determinant formula that settles two residue classes in the Rudin–Shapiro problem.