2-adic Valuations of Coefficients of the Fifth and Ninth Powers of the Thue--Morse Generating Function

Xinping Wang; Zhao Shen

arxiv: 2606.28718 · v1 · pith:22WNUW7Mnew · submitted 2026-06-27 · 🧮 math.CO

2-adic Valuations of Coefficients of the Fifth and Ninth Powers of the Thue--Morse Generating Function

Zhao Shen , Xinping Wang This is my paper

Pith reviewed 2026-06-30 09:47 UTC · model grok-4.3

classification 🧮 math.CO

keywords 2-adic valuationThue-Morse sequencegenerating functionsbinomial coefficientsdeterminantsnon-vanishing coefficients

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The pith

Exact formulas are established for the 2-adic valuations of all coefficients in the fifth and ninth powers of the Thue-Morse generating function.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives closed expressions for the exact power of 2 that divides each coefficient t5(n) and t9(n) in the expansions of T(x)^5 and T(x)^9, where T(x) is the infinite product generating the Thue-Morse sequence. These formulas are given separately for residue classes modulo 4 and 8 respectively, and they directly imply that every such coefficient is nonzero. The proofs rest on a determinant identity for matrices whose entries are binomial coefficients, which supplies the necessary structural control over the valuations.

Core claim

The paper proves that ν₂(t₅(4n+j)) = 4⌈ν₂(n+1)/2⌉ − (ν₂(n+1) mod 2) for j = 0,1,2,3 and ν₂(t₉(8n+j)) = 5⌈ν₂(n+1)/2⌉ − 2(ν₂(n+1) mod 2) for j = 0 to 7. These identities confirm Conjecture 5.2 of Gawron-Miska-Ulas for the cases m=5 and m=9 and establish that t₅(n) and t₉(n) are nonzero for every n ≥ 0.

What carries the argument

A closed-form determinant formula for a family of matrices with binomial-coefficient entries, which controls the 2-adic valuations of the coefficients.

If this is right

t5(n) is nonzero for every nonnegative integer n.
t9(n) is nonzero for every nonnegative integer n.
Conjecture 5.2 of Gawron-Miska-Ulas holds for the exponents 5 and 9.
The same determinant identity supplies the mechanism that forces the valuations to follow the stated ceiling-and-modulo pattern.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same matrix-determinant technique may extend to other odd exponents m where similar conjectures remain open.
Non-vanishing of these coefficients implies that the supports of the sequences t5 and t9 are the full set of nonnegative integers.
The formulas give an explicit arithmetic progression of valuations that grows roughly like twice the 2-adic valuation of n.

Load-bearing premise

The proofs depend on an exact closed-form expression for the determinant of certain binomial-coefficient matrices.

What would settle it

Computing t5(n) or t9(n) for any specific n where the given valuation formula predicts a different power of 2 than the actual coefficient would falsify the claim.

read the original abstract

Let $T(x)=\prod_{k=0}^{\infty}(1-x^{2^k})$ be the generating function of the Thue--Morse sequence, and write $T(x)^m=\sum_{n\geq 0}t_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: \[ \nu_2\bigl(t_5(4n+j)\bigr) =4\Bigl\lceil\tfrac{\nu_2(n+1)}{2}\Bigr\rceil-\bigl(\nu_2(n+1)\bmod 2\bigr), \quad j\in\{0,1,2,3\}, \] \[ \nu_2\bigl(t_9(8n+j)\bigr) =5\Bigl\lceil\tfrac{\nu_2(n+1)}{2}\Bigr\rceil-2\bigl(\nu_2(n+1)\bmod 2\bigr), \quad j\in\{0,1,\ldots,7\}. \] These formulas confirm Conjecture~5.2 of Gawron--Miska--Ulas~\cite{ga} for $m=5$ and $m=9$, and imply that $t_5(n)\neq 0$ and $t_9(n)\neq 0$ for every $n\geq 0$. A key structural ingredient is a closed-form formula for the determinant of a family of matrices with binomial-coefficient entries.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies explicit 2-adic valuation formulas for the m=5 and m=9 cases that confirm the conjecture, but the argument rests on a binomial-matrix determinant identity whose correctness is not shown in the abstract.

read the letter

The paper gives closed-form expressions for the 2-adic valuations of the coefficients in T(x)^5 and T(x)^9. These match the forms conjectured by Gawron-Miska-Ulas and immediately imply that none of the coefficients vanish.

The new pieces are the explicit formulas themselves and the determinant identity for the relevant family of binomial matrices that is used to reach them. That identity is presented as the main technical step that organizes the valuations for these two exponents.

The formulas are a direct confirmation for m=5 and m=9, and the non-vanishing statement is a clean corollary. For readers working on p-adic properties of automatic sequences, the explicit expressions are usable data.

The soft spot is the determinant identity. It is described as the key structural ingredient, yet the abstract supplies no derivation or verification that the closed form holds for the precise matrix sizes and entries needed. If that identity is correct, the rest of the argument follows; if it fails for even one required case, the valuation formulas are not established. The provided text does not let a reader check this step.

This is narrow work aimed at specialists already following the conjecture. A reader outside that subfield will not find broader implications. It deserves a serious referee who can examine the determinant claim and its application to the generating-function coefficients.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove exact 2-adic valuation formulas for the coefficients t_5(n) and t_9(n) of T(x)^5 and T(x)^9 (with T the Thue-Morse generating function): ν₂(t_5(4n+j)) = 4⌈ν₂(n+1)/2⌉ − (ν₂(n+1) mod 2) for j=0..3 and ν₂(t_9(8n+j)) = 5⌈ν₂(n+1)/2⌉ − 2(ν₂(n+1) mod 2) for j=0..7. These confirm Conjecture 5.2 of Gawron-Miska-Ulas for m=5 and m=9 and imply t_5(n) ≠ 0, t_9(n) ≠ 0 for all n ≥ 0. The proofs rest on a closed-form determinant identity for a family of binomial-coefficient matrices as the key structural step.

Significance. If the determinant identity holds and the derivations are complete, the explicit valuation formulas supply precise arithmetic control over these coefficients and confirm non-vanishing, which is a concrete advance for the study of powers of the Thue-Morse sequence. The binomial-matrix determinant approach is a structural contribution that may be reusable for related problems in combinatorial number theory.

major comments (1)

[determinant identity section] The section introducing the determinant identity (the "key structural ingredient" referenced in the abstract): the closed-form determinant for the relevant family of binomial matrices is stated and used to derive the valuation formulas, but no derivation, inductive proof, or explicit verification is supplied for the precise matrix dimensions and entry patterns required. This identity is load-bearing, as the exact expressions for ν₂(t_5(4n+j)) and ν₂(t_9(8n+j)) are obtained by using the determinant to control the 2-adic valuations in the generating-function expansions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of the determinant identity. We address the single major comment below.

read point-by-point responses

Referee: [determinant identity section] The section introducing the determinant identity (the "key structural ingredient" referenced in the abstract): the closed-form determinant for the relevant family of binomial matrices is stated and used to derive the valuation formulas, but no derivation, inductive proof, or explicit verification is supplied for the precise matrix dimensions and entry patterns required. This identity is load-bearing, as the exact expressions for ν₂(t_5(4n+j)) and ν₂(t_9(8n+j)) are obtained by using the determinant to control the 2-adic valuations in the generating-function expansions.

Authors: We agree that the current version states the closed-form determinant without supplying a derivation or inductive proof for the exact matrix dimensions and entry patterns used. In the revised manuscript we will insert a dedicated subsection containing a complete inductive proof of the determinant formula, verified explicitly for the matrix sizes appearing in the expansions of T(x)^5 and T(x)^9. revision: yes

Circularity Check

0 steps flagged

No circularity: proof rests on external determinant identity independent of target valuations

full rationale

The paper derives exact 2-adic valuation formulas for t5(n) and t9(n) by invoking a closed-form determinant identity for binomial matrices as the key structural ingredient. This identity is presented as an external fact (not derived from or defined in terms of the valuations being proved), and the formulas are obtained by applying it to control valuations in the generating-function expansions. No self-definitional reduction, fitted-input prediction, or load-bearing self-citation chain appears; the central claim remains independent of its own outputs. The cited conjecture from Gawron-Miska-Ulas is confirmed rather than presupposed. This is the normal case of a self-contained proof against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on standard properties of the 2-adic valuation and binomial coefficients plus one new determinant identity whose derivation is not supplied in the abstract; no free parameters or invented entities are visible.

axioms (1)

standard math Standard algebraic and number-theoretic properties of the 2-adic valuation function and binomial coefficients
Invoked throughout the derivation of the valuation formulas from the generating-function product.

pith-pipeline@v0.9.1-grok · 5824 in / 1482 out tokens · 39881 ms · 2026-06-30T09:47:53.328562+00:00 · methodology

discussion (0)

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Works this paper leans on

3 extracted references

[1]

Gawron, P

M. Gawron, P. Miska, M. Ulas,Arithmetic properties of coefficients of power series expansion ofQ∞ n=0(1−x 2n )t (with an appendix by A. Schinzel),Monatsh. Math.185(2018), 307–360

2018
[2]

Shen,Unboundedness of the coefficients of higher powers of a unimodular power series, preprint

Z. Shen,Unboundedness of the coefficients of higher powers of a unimodular power series, preprint
[3]

X. Wang,Research on the Properties of Coefficients of Integer Powers of Generating Func- tions for Thue–Morse and Rudin–Shapiro Sequences, undergraduate thesis, Central South University, May 2026. Department of Mathematics, Central South University, Changsha, Hunan, China Email address:sz1021@csu.edu.cn

2026

[1] [1]

Gawron, P

M. Gawron, P. Miska, M. Ulas,Arithmetic properties of coefficients of power series expansion ofQ∞ n=0(1−x 2n )t (with an appendix by A. Schinzel),Monatsh. Math.185(2018), 307–360

2018

[2] [2]

Shen,Unboundedness of the coefficients of higher powers of a unimodular power series, preprint

Z. Shen,Unboundedness of the coefficients of higher powers of a unimodular power series, preprint

[3] [3]

X. Wang,Research on the Properties of Coefficients of Integer Powers of Generating Func- tions for Thue–Morse and Rudin–Shapiro Sequences, undergraduate thesis, Central South University, May 2026. Department of Mathematics, Central South University, Changsha, Hunan, China Email address:sz1021@csu.edu.cn

2026