Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$

Mark L. Lewis; Shannon M. Tefft

REVIEW 3 major objections 6 minor 21 references

Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$

T0 review · 3 major / 6 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read For the Ducci map on $\mathbb{Z}_p^p$ with $p$ an odd prime, the paper classifies every possible cycle length: $1$ for the zero tuple, the order of $2$ modulo $p$ for constant tuples, and the maximum $p$ times that order for every other…

desk verdict Theorem 4 is a correct and clean period trichotomy for n=m=p; the paper is solid and deserves review, with the self-cited lemmas and code made explicit. read the letter →

arxiv 2502.03348 v1 pith:NM75HPLP submitted 2025-02-05 math.NT math.GR

Mark L. Lewis , Shannon M. Tefft This is my paper · ORCID

classification math.NTmath.GR MSC 20D6011B8311B50

keywords Duccisequenceperiodmodulararithmeticbasicn-NumberGamebinomialcoefficientscoefficientmatrix

topics The Collatz Conjecture

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The Ducci map sends a cyclic tuple of residues modulo $m$ to the tuple of sums of neighboring entries modulo $m$. Every tuple eventually enters a cycle, and this paper asks which cycle lengths can actually occur. For the square prime case $n=m=p$, the answer is complete: the only periods are $1$ for the zero tuple, $\delta$ (the multiplicative order of $2$ modulo $p$) for nonzero constant tuples, and the full period $p\delta$ for every other tuple. For three coordinates with odd prime modulus, it proves that any tuple that is neither zero, constant, nor sum-zero has the maximum period $P_m(3)$. These results give full period spectra for natural infinite families of Ducci processes.

What carries the argument

The workhorse is the family of coefficients $a_{r,s}$: the coefficient of $x_s$ in the first coordinate of $D^r(x_1,\ldots,x_n)$. For $r<n$ these are binomial coefficients $\binom{r}{s-1}$, and they satisfy a convolution identity $a_{r+t,s}=\sum_i a_{t,i}a_{r,s-i+1}$. When $n=p$ is prime, the binomial coefficients collapse modulo $p$: $a_{p,s}\equiv 0$ for $s\neq 1$ and $a_{p,1}=2$, so $D^p(\mathbf{u})=2\mathbf{u}$ for every tuple. This identifies the full period as $p\delta$ and reduces the search for shorter periods to tuples fixed by $D^{p-1}$; the proof then shows $D^{p-1}(\mathbf{u})=\mathbf{u}$ forces $\mathbf{u}$ to be zero or constant by inverting the alternating-sign coefficient matrix of Lemma 11, whose determinant is $\pm 1$.

What would settle it

Enumerate all $5^5=3125$ tuples of $\mathbb{Z}_5^5$ and compute their periods under $D$. Theorem 4 predicts exactly one tuple with period $1$ (the zero tuple), four with period $4$ (the nonzero constant tuples), and all $3120$ remaining tuples with period $20$. Finding any nonconstant nonzero tuple that returns to itself in fewer than $20$ steps would refute the classification.

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Extended reading notes

Core claim

The central claim is Theorem 4: if $n=m=p$ is an odd prime and $\delta$ is the order of $2$ modulo $p$, then the only possible values of $\mathrm{Per}(\mathbf{u})$ on $\mathbb{Z}_p^p$ are $1$, $\delta$, and $p\delta$. The zero tuple alone has period $1$; a tuple $(x,\ldots,x)$ with $x\neq 0$ has period $\delta$; every other tuple has period $P_p(p)=p\delta$. The paper also establishes Theorem 3 for $n=3$: for odd prime $m$, every tuple $\mathbf{u}=(x_1,x_2,x_3)$ whose entries are not all equal and whose sum is not $0$ modulo $m$ has period $P_m(3)$, while nonzero sum-zero tuples have period at most $6$ by Lemma 2.

Load-bearing premise

The proof of Theorem 3 and the no-preperiod steps in Theorem 4 assume the companion result that for odd prime modulus every tuple already lies on its Ducci cycle rather than having a preperiod; without that, the equations $D^d(\mathbf{u})=\mathbf{u}$ used throughout the argument are not guaranteed.

Editorial extensions

If this is right

For $n=m=p$, the maximum period is exactly $p\delta$, and it is achieved by every tuple except the zero tuple and the nonzero constant tuples; in particular there is no intermediate period.
For $n=3$ with odd prime modulus, the sum condition $x_1+x_2+x_3\equiv 0$ is the only source of short periods among nonconstant tuples, and those tuples have period $6$ when nonzero.
Uniformly constant tuples always have period equal to the order of $2$ modulo the odd part of $m$, so they provide a recurring short period in every dimension.
If $m_1\mid m$, every period realized in $\mathbb{Z}_{m_1}^n$ is also realized in $\mathbb{Z}_m^n$, so the set of possible periods is monotone in the modulus.
For the prime pairs tabulated in Figures 3 through 5, the exceptional short periods are rare and, in several cases, the tuples of a given exceptional period form orbits under a dihedral or Frobenius symmetry group.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The methods point toward a conjecture the authors do not make: for $n=m=p^k$ with $p$ an odd prime, the period spectrum should be governed by the order of $2$ modulo $p^k$ and by lifts of the prime-power factors, and Lemma 6 provides the divisibility half of such a classification.
The observed stabilizers for exceptional periods (dihedral groups of order $2n$ and a Frobenius group of order $21$) suggest that counting tuples with a given exceptional period can be reduced to counting orbits of the symmetric group on solution sets of $D^d(\mathbf{u})=\mathbf{u}$, a count the paper does not attempt.
For a fixed dimension $n$, the tables show the sum condition is not by itself a period shortcut; a general theorem would need to read the coefficients $a_{d,s}$ modulo $m$, since the sums of coordinates alone cannot predict which short period will appear.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear Colleague,

The thing to know: Theorem 4 is the real contribution—a complete period trichotomy for n=m=p prime—and it is correct. Theorem 3 gives the n=3 classification and is also sound, with a real but replaceable dependence on prior work. This is solid subfield-level progress, not a breakthrough.

What is new: the complete set of periods for n=m=p and the n=3, m odd prime classification appear in no cited prior paper. The proof strategy is clean: over F_p^p, D^p = 2I, so periods divide pδ; a divisor analysis plus invertibility of the (p−1)×(p−1) matrix in Lemma 11 leaves only 1, δ, pδ. The determinant computation is correct, and the fixed-point argument rules out intermediate periods.

The n=3 theorem also holds up. It does invoke Theorem 7 of [13] to get L_m(3)=0 for m odd prime before writing equations (4.1)–(4.3). That is a genuine black-box dependency, but the stress-test note is right that the fact is true and can be derived in a couple lines from D = I + H, so it is not a fatal gap. Still, the paper should either cite precisely or include the short proof. There is a similar, less serious reliance on Corollary 6 of [12] in the proof of Theorem 4; also easy to prove from binomial coefficients.

The weakest section is 5. The computational census is presented as evidence, but the MATLAB code and data are not shipped, and the search stops at periods over 15 million. That is a real limitation, but it is not load-bearing for Theorems 3 and 4. For a paper that uses computation to support patterns, shipping code would be a meaningful improvement.

The self-citation pattern is fair; the cited facts are needed and appear correct. I did not find inflated claims or circularity.

Who it is for: people working on Ducci sequences or discrete maps over finite abelian groups. The paper should be refereed, not desk-rejected. I would ask for the black-box lemmas to be proved or quoted exactly, and for the computational code to be made available. A minor revision should be enough.

Referee Report

3 major / 6 minor

Summary. The paper studies the period structure of the Ducci map D(x_1,...,x_n)=(x_1+x_2,...,x_n+x_1) on Z_m^n. The main results are Theorem 3, which states that for n=3 and m an odd prime, every tuple that satisfies neither the sum condition nor the uniformity condition has maximal period P_m(3), and Theorem 4, which states that for n=m=p an odd prime, the only possible periods are 1 (for the zero tuple), δ (the multiplicative order of 2 modulo p, for uniform nonzero tuples), and pδ (for all other tuples). The paper also contains auxiliary lemmas about uniform tuples, sum-condition tuples, and period lifting under divisors of the modulus, as well as computational tables for small prime n,m and a discussion of symmetry groups of exceptional periods.

Significance. If the results are correct, the paper gives a satisfyingly complete picture of possible periods in the n=3 odd-prime case and in the n=m=p case, with the latter being a sharp three-value classification. The methods are elementary but the coefficient machinery a_{r,s} is useful, and the computational tables provide concrete data that could guide future work. The paper is modest in scope but represents a solid contribution to the Ducci-sequence literature. I explicitly note that Theorem 4's classification is falsifiable and the proof is checkable; the main obstacles are the false statement of Lemma 5 for composite moduli and the reliance on same-author preprints for load-bearing facts, both of which are fixable.

major comments (3)

[Section 2, Lemma 5] Lemma 5 as stated is false for composite odd m_1. For example, take m=9 (so l=0, m_1=9) and x=3. The order of 2 modulo 9 is δ=6, but 2^2·3 ≡ 3 mod 9, so D^2(3,3,...,3)=(3,3,...,3) and the period is 2, not 6. The proof implicitly treats the condition 2^r x ≡ x mod m_1 as equivalent to 2^r ≡ 1 mod m_1, which is only valid when gcd(x,m_1)=1. The lemma should either assume gcd(x,m_1)=1 or restrict m_1 to be prime; the statement and proof must be corrected. This does not invalidate Theorem 4 because there m_1=p is prime and every nonzero x is invertible modulo p, but the lemma as written is a serious mathematical error.
[Section 6, proof of Theorem 4] The derivation of P_p(p)=pδ contains a garbled citation: the text says 'because of Corollary 6 in [12] a_{p,1}=2 and a_{p,s}=binom(p,s-1)', but binom(p,0)=1 for s=1, which contradicts a_{p,1}=2. Moreover, the binomial formula a_{r,s}=binom(r,s-1) in Theorem 5 of [12] is stated only for 0≤r<n, whereas here r=n=p, so the formula does not apply as written. The intended conclusion D^p(u)=2u is correct and follows directly from D=I+H and the binomial theorem over F_p: D^p=(I+H)^p≡I^p+H^p=2I mod p. Please replace the incorrect citation with this direct proof.
[Theorem 3 and Theorem 4] The proofs of Theorem 3 and Theorem 4 depend on load-bearing facts quoted from the same authors' unpublished preprints: Theorem 7 of [13] (L_m(3)=0 for odd prime m) is used in the proof of Theorem 3 to justify equations (4.1)-(4.3), and Theorem 2 of [13] (L_p(p)=0) is used in Theorem 4, along with Corollary 6 of [12]. Since [12] and [13] are listed as 'Submitted for Publication' and are not yet peer-reviewed, the paper should either prove these facts (for example, L_m(3)=0 for odd prime m follows because D=I+H and det(I+H)=2 is invertible over F_m) or cite a published source. This is not a stylistic request: without L_m(3)=0, the equations D^d(u)=u used in the proof of Theorem 3 would not follow for tuples with a nonzero preperiod.

minor comments (6)

[Lemma 2(5)] In the proof of part (5), the sentence 'if x_1,x_2,x_3 ∈ {0,m/2}, then Per(u)=3' should explicitly exclude the all-equal tuple (m/2,m/2,m/2), which has D(m/2,m/2,m/2)=(0,0,0) and period 1, not 3.
[Theorem 3 proof] The case m=3 is dismissed with 'if you plot out the Ducci sequences for all 27 tuples'; for a formal proof, please include a short table or a direct argument showing that only uniform tuples have period less than 6.
[Section 5] The computational claims summarized in Figures 3-5 are not accompanied by the MATLAB code or a precise description of the algorithm beyond the n=5,m=7 example; for reproducibility, please include the program as an ancillary file or provide pseudocode.
[Lemma 8] The induction proof for the residues 2, 3, 4, 5 modulo 6 is only sketched with 'We can repeat this pattern'; please expand this step so the reader can verify the claimed congruences without reconstructing the induction.
[Section 6] The line 'We can now prove Theorem ??' before the proof of Theorem 4 contains a placeholder and should read 'Theorem 4'.
[Section 2, Lemma 5] In the proof of Lemma 5, the phrase 'Since δ is the smallest value where this can happen' is only true when x is invertible modulo m_1; the corrected statement and proof should make the coprimality assumption explicit.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: central results are self-contained; self-citations are replaceable by elementary arguments.

full rationale

The paper's definitions of Per(u), P_m(n), and L_m(n) are fixed independently, and no parameter is fitted to data and then renamed a prediction. The core of Theorem 4 is proved from first principles: D^p = 2I follows from the binomial coefficient identity in Z_p, the divisor analysis of periods uses only the multiplicative order of 2 and the fact that D^p acts as 2I, and the final fixed-point system D^{p-1}(u)=u is solved by the explicitly proved Lemma 11. The only self-citations that appear load-bearing in the written proofs are the assertions L_m(3)=0 and L_p(p)=0 taken from the authors' earlier work [13]. These are not circular: they are external mathematical facts about the Ducci map being automorphic for odd modulus, verifiable by determinant or invertibility arguments (e.g., the circulant matrix 1+x is coprime to x^p-1 over F_p for odd p), and they do not assume the period classifications being proven. Likewise, the coefficient facts imported from [12] are standard binomial-coefficient identities. Thus the central claims do not reduce to their own inputs, and the self-citations, while present, do not constitute circular reasoning.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central proofs are largely self-contained but call on prior Ducci theorems, several from the same authors' preprints, as black boxes. No numerical parameters are fitted and no new entities are postulated.

assumptions (6)

domain assumption For all u in Z_m^n, Per(u) divides P_m(n) and Len(u) <= L_m(n) (Lemma 1 of [1]).
Used throughout to restrict possible periods to divisors of the basic Ducci period; imported from Breuer 1999.
domain assumption Basic properties of the coefficients a_{r,s}: recurrence a_{r,s}=a_{r-1,s}+a_{r-1,s-1}, binomial values for r<n, and convolution formula (Theorem 5 of [12]).
Foundation for Corollary 7, Lemma 8, and the fixed-point equations in Sections 3-6; from the same authors' submitted preprint.
domain assumption For n=p prime and m=p, D^p(u)=2u for all u (Corollary 6 of [12]).
Used in Theorem 4 to compute P_p(p)=p*delta and to exclude intermediate multiples of p as periods.
domain assumption L_m(3)=0 for m an odd prime (Theorem 7 of [13]).
Used in Theorem 3 to assert D^{Per(u)}(u)=u, so the fixed-point linear system (4.1)-(4.3) applies.
domain assumption L_p(p)=0 for p an odd prime (Theorem 2 of [13]).
Used in Theorem 4 to argue only the zero tuple has period 1 because zero has a unique predecessor.
standard math Binomial congruence binom(p-1,s-1) = (-1)^{s-1} mod p for prime p.
Used in Theorem 4 to compute coefficients of D^{p-1}; proof cited to Lemma 4 of [14], but it is a standard fact.

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Pith. "Pith review of Values of Ducci Periods for Sequences on $\mathbb{Z}_m^n$." pith.science (2026). https://pith.science/paper/NM75HPLP

@misc{pith2026250203348,
  author       = {Pith},
  title        = {Pith review of: Values of Ducci Periods for Sequences on $\mathbbZ_m^n$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NM75HPLP}},
  note         = {Machine review of arXiv:2502.03348}
}

abstract

Let $D: \mathbb{Z}_m^n \to \mathbb{Z}_m^n$ be defined so that \[D(x_1, x_2, ..., x_n)=(x_1+x_2 \; \text{mod} \; m, x_2+x_3 \; \text{mod} \; m, ..., x_n+x_1 \; \text{mod} \; m).\] We call $D$ the Ducci function and the sequence $\{D^{\alpha}(\mathbf{u})\}_{\alpha=0}^{\infty}$ the Ducci sequence of $\mathbf{u}$ for $\mathbf{u} \in \mathbb{Z}_m^n$. Every Ducci sequence enters a cycle, so we can let $\text{Per}(\mathbf{u})$ be the number of tuples in the Ducci cycle of $\mathbf{u}$, or the period of $\mathbf{u}$. In this paper, we will look at what different possible values of $\text{Per}(\mathbf{u})$ we can have and some conditions that if $\mathbf{u}$ meets at least one of them, $\mathbf{u}$ will generate a period smaller than the maximum period.

Figures

Figures reproduced from arXiv: 2502.03348 by the authors.

**Figure 1.** Transition Graph for Z 3 4 This therefore opens up one other possible period when m is not a power of 2. Note that this may not always be a distinct period from Pm(n). For example, P11(5) = 10 and the multiplicative order of 2 mod 11 is also 10. Since 11 is prime, all tuples that satisfy the uniformity condition in Z 5 11 still generate the maximum period P11(5). To discuss the next type of possible period that can … view at source ↗

**Figure 2.** Part of Transition Graph for Z 3 m, m Even If Per(u) = 1, 2, then u = D2 (u) = H(u) implies x1 = x2 = x3, which is a contradiction. If Per(u) = 3, then u = D3 (u) = D(x2, x3, x1) implies x2 + x3 ≡ x1 mod m x1 + x3 ≡ x2 mod m x1 + x2 ≡ x3 mod m. Plugging the third equation into the first yields 2x2 + x1 ≡ x1 mod m, or that 2x2 ≡ 0 mod m. Similarly, 2x3 ≡ 0 mod m and 2x1 ≡ 0 mod m. If m is odd, then this only happens … view at source ↗

**Figure 3.** n, m prime, three possible periods We once more multiply Equations (4.1), (4.2), and (4.3) by 3 and use these equivalencies to yield 2x1 − x2 + 2x3 ≡ 3x1 mod m 2x1 + 2x2 − x3 ≡ 3x2 mod m −x1 + 2x2 + 2x3 ≡ 3x3 mod m. However, these are Equations (4.5), (4.6), and (4.7) after moving all of the xi to the left side, which still results in x1 ≡ x2 mod m ≡ x3 mod m and a contradiction. Therefore, the only possibility is t… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: n, m Prime, four possible periods m. Observing Figures 3 and 4, the value of Pm(n) typically increases as n, m increase. As a result, for many n, m, Pm(n) becomes too large for MATLAB to find, preventing us from testing these cases. Our current program is currently set…

**Figure 5.** Figure 5: n, m prime, Exceptions x1 = j for 2 ≤ j ≤ 6 and find that xi = j for all 1 ≤ i ≤ 5. In other words, only ~0 and those tuples satisfying the uniformity condition, which we already know have period 3, are solutions to this system of equations. Therefore, 120 is not a per…

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Reference graph

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.

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Ducci on $\mathbb{Z}_m^n$ and the Maximum Length for $n$ Odd

Lewis, M.L. & Teﬀt, S.M. (2024). Ducci on Zn m and the Maximum Length for n Odd. Sub- mitted for Publication. ¡arXiv:2403.05319¿

work page Pith review arXiv 2024

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The Period of Ducci Cycles on $\mathbb{Z}_{2^l}$ for Tuples of Length $2^k$

Lewis, M.L. & Teﬀt, S.M. (2024). The Period of Ducci Cycl es on Z2l for Tuples of Length 2k. Submitted for Publication. <arXiv: 2401.17502 >

work page Pith review arXiv 2024

[3] [1]

Breuer, F. (1999). Ducci Sequences Over Abelian Groups. Communications in Algebra, 27(12), 5999-6013

work page 1999

[4] [2]

Breuer, F. (2010). Ducci Sequences and Cyclotomic Field s. Journal of Diﬀerence Equations and Applications, 16(7) , 847-862

work page 2010

[5] [3]

& Merzel, J

Brown, R. & Merzel, J. (2003). Limiting Behavior in Ducci Sequences. Periodica Mathemat- ica Hungarica, 47(1-2) , 45-50

work page 2003

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& Merzel, J

Brown, R. & Merzel, J. (2007). The Length of Ducci’s Four N umber Game Rocky Mountain Journal of Mathematics, 37(1) , 45-65. V ALUES OF DUCCI PERIODS FOR SEQUENCES ON Zn m 21

work page 2007

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(1978) Circles o f Numbers

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work page 1978

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Chamberland, M. (2003). Unbounded Ducci Sequences. Journal of Diﬀerence Equations and Applications, 9(10) , 887-895

work page 2003