Colored base-3 partitions, sequences of polynomials, and perfect numbers

Karl Dilcher; Larry Ericksen

Colored base-3 partitions, sequences of polynomials, and perfect numbers

T0 review · 0 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Every even perfect number appears as a colored base-3 partition count

desk verdict A correct and elegant combinatorial observation—the perfect-number counts are real but explanatory—with localized but genuine errors in Section 5. read the letter →

arxiv 2509.03147 v1 pith:DED7RYI2 submitted 2025-09-03 math.CO math.NT

Karl Dilcher , Larry Ericksen This is my paper · ORCID

classification math.COmath.NT MSC 11P8111B3711B83

keywords base-3partitionscoloredrestrictedperfectnumbersChebyshevpolynomialsrecurrencerelationszerosofMersenneprimes

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

This paper shows that the counting function for colored base-3 partitions, restricted to at most one overline, one tilde, and two unmarked copies of each power of 3, contains every even perfect number as a subsequence. The authors define a four-variable polynomial analogue S(n;Z) whose coefficients track how many parts carry each kind of mark, then isolate the subsequence Q_n = S((3^n−3)/2;Z). After setting all variables to 1, this subsequence satisfies q_n = 6q_{n−1} − 8q_{n−2}, with closed form q_n = 2^{n−1}(2^n−1). By the Euclid–Euler theorem, q_n is a perfect number exactly when n is prime and 2^n−1 is prime, and every even perfect number arises this way. The same machinery connects the polynomials to Chebyshev polynomials and yields explicit combinatorial counts for marked and unmarked parts.

What carries the argument

The load-bearing object is the polynomial sequence S(n;Z), defined by the generating function ∏_{j≥0} (1+wq^{3^j})(1+xq^{3^j})(1+yq^{3^j}+zq^{2·3^j}); setting w=x=y=z=1 makes its coefficients count the restricted colored base-3 partitions. The proof runs through the recurrence system (2.4)–(2.6), taken from the authors' earlier work. For the subsequences Q_n(Z)=S((3^n−3)/2;Z) and R_n(Z)=S((3^n−1)/2;Z), the recurrence condenses to Q_n = W1 Q_{n−1} − W2 Q_{n−2} with W1 = wxy+wz+xz+w+x+y and W2 = w^2xy+w^2z+wx^2y+wxy^2+wxz+wyz+x^2z+xyz. At Z=(1,1,1,1) this becomes q_n = 6q_{n−1} − 8q_{n−2}; its Binet solution q_n = 2^{n−1}(2^n−1) is the perfect-number formula. A Chebyshev representation Q_{n+1}

What would settle it

Enumerate all restricted colored base-3 partitions of 39 = (3^4−3)/2 under condition (1.1) by brute force; the formula predicts exactly 120. A different count would refute the central recurrence and the perfect-number connection.

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Extended reading notes

Core claim

The central claim is Corollary 3.2: for every n ≥ 1, exactly 2^{n−1}(2^n−1) restricted colored base-3 partitions of (3^n−3)/2 satisfy condition (1.1). In particular, when n is prime and 2^n−1 is also prime, that count is a perfect number; conversely, by the Euclid–Euler characterization, every even perfect number is obtained by choosing n equal to a Mersenne-prime exponent. The paper explains the mechanism: the subsequence Q_n obeys a second-order linear recurrence whose characteristic polynomial factors as (t−4)(t−2), and the Binet formula is exactly the Euclid–Euler product. The authors further identify the companion subsequence R_n = S((3^n−1)/2;Z), which gives 2^{n−1}(2^n+1), and develop

Load-bearing premise

The argument depends on the three recurrence relations (2.4)–(2.6), imported from the authors' earlier paper without re-derivation, and on the informal identification of the generating-function coefficients with the number of restricted partitions; if either gives way, the perfect-number formula and its corollaries collapse.

Editorial extensions

If this is right

The restricted base-3 partition counting sequence S(n) hits 6, 28, 496, 8128, ... at n = 3, 12, 120, 1092, reproducing every even perfect number.
The companion sequence r_m = 2^{m−1}(2^m+1) counts partitions of (3^m−1)/2; for m = 2^k with k = 0..4 these values connect to Fermat primes and constructible regular polygons.
The generating functions of Q_n and R_n are rational with denominator 1 − W1 q + W2 q^2, so all these counts and polynomials satisfy the same second-order recurrence, giving fast computation independent of partition enumeration.
For Z = (1,1,z,1), the polynomials have closed forms ((z+3)^n − (z+1)^n)/2 and ((z+3)^n + (z+1)^n)/2; their coefficients count partitions by number of single unmarked parts, and all zeros lie on the vertical line Re z = −2.
For the three single-variable specializations studied, the Q-polynomials are divisibility sequences: if m divides n then Q_m^(j)(z) divides Q_n^(j)(z).

Reading between the lines

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

The paper does not supply a bijection between the 2^{n−1}(2^n−1) partitions and pairs (a,b) with a ≤ 2^{n−1}, b ≤ 2^n−1; finding such a bijection would make the perfect-number product visible combinatorially rather than through Binet's formula.
The polynomial Q^(1)_n(z) = ((z+3)^n − (z+1)^n)/2 can be read as a refinement of the perfect-number count, graded by the number of unmarked parts; its factorization properties may suggest polynomial analogues of perfect numbers.
The zero-location proofs for the three worked specializations all follow the same Chebyshev argument; the six additional cases listed in Section 5.3, such as Z = (z,z,1,1) and Z = (1,1,z,z^2), should yield explicit zero curves by the same method, though the paper only states their combinatorial meaning.
If the recurrence system (2.4)–(2.6) were derived directly from a combinatorial splitting of base-3 partitions into blocks, the perfect-number connection would become self-contained and might generalize to other bases with analogous restriction types.

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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

Here's my take. The central claim (Corollary 3.2) is right: the restricted colored base-3 partition counts at indices (3^n−3)/2 are exactly 2^{n−1}(2^n−1), so every even perfect number appears as such a count. This is a genuine surprise in the model, and the proof is clean. I verified the induction in Proposition 3.1 and the characteristic-root derivation; both hold. The recurrences come from [1], but the stress-test note is correct that they follow immediately by comparing coefficients in the generating function (2.1), so the import is not a load-bearing gap. The Chebyshev transfer in Propositions 3.5 and 4.2 is elegant and correct.

Where the paper is soft: Section 5. Proposition 5.3 is false as stated because the zero at z=0 of multiplicity n−1 is omitted. Actually, from the recurrence (5.3), Q_n^(2)(0)=0 for all n≥2, so the 'if and only if' claim fails. There is also an apparent factor-2 inconsistency between (5.2)/(5.7) and the correct substitution from (3.14), which should be 3/(4√2)=3√2/8, not 3√2/4; this propagates into the proof of Proposition 5.3. And Section 5.3 lists six additional cases with zero proofs. These defects are peripheral—they do not touch Corollary 3.2 or Section 4—but they need fixing before publication.

The perfect-number observation is explanatory rather than revelatory: once you have the closed form, the Euclid–Euler theorem does the rest. That is a feature, not a bug, but it caps the significance. The paper is best read as the latest in the authors' established program, so the novelty is incremental. That said, the combinatorial interpretations and the zero-distribution results are new and well motivated.

Who gets value: people working on restricted b-ary partitions, polynomial analogues, or Chebyshev-linked polynomial families. It deserves a serious referee; I would accept it with revision. The central result is sound enough to fix the Section 5 issues without redoing the paper.

Referee Report

0 major / 5 minor

Summary. This paper studies a restricted colored base-3 partition function S(n) defined by the type (1,1,2) condition (1.1). It introduces a four-variable polynomial generating function S(n;Z) in (2.1), records recurrences (2.4)-(2.6), and then focuses on the subsequences Q_n(Z)=S((3^n-3)/2;Z) and R_n(Z)=S((3^n-1)/2;Z). The main result is Corollary 3.2: for Z=(1,1,1,1), S((3^n-3)/2)=2^{n-1}(2^n-1), so that indices (3^n-3)/2 for Mersenne-prime n count even perfect numbers; conversely all even perfect numbers occur this way. The proof uses an induction for the second-order recurrence q_n=6q_{n-1}-8q_{n-2} and Binet formulas. The remainder of the paper derives Chebyshev-polynomial expressions and zero-location results for specializations Z1=(1,1,z,1), Z2=(z,z,z,z^2), Z3=(1,1,z,z), and sketches six further cases.

Significance. If correct, the paper provides a new combinatorial interpretation of the even perfect numbers as counts of restricted colored base-3 partitions. I checked the central chain: Proposition 3.1's induction is valid, the characteristic roots give (3.9)-(3.10), and the Euclid-Euler step is immediate. The Chebyshev connections (3.14)-(3.15) are clean and lead to explicit zero locations. The paper's strengths are the transparent linear-recurrence proof and the surprising subsequence identification; the defects I found are local and correctable.

minor comments (5)

[5.1, Proposition 5.3] Proposition 5.3 is false as stated: for every n ≥ 2 the polynomial Q(2)_n(z) has z = 0 as a zero (lowest exponent n−1 by Prop. 5.1), yet z=0 does not lie on the segments |Im z| > 1/3. The proof's equivalence (5.7) is valid only for z ≠ 0. The proposition should be corrected to state that all nonzero zeros lie on those two arcs of the unit circle, with z=0 as an additional zero. This does not affect Corollary 3.2, but it needs fixing.
[2, Propositions 2.2–2.4] The recurrences (2.4)–(2.6), which are the engine for all later results, are quoted from [1, Thm. 4.3] without proof. Since they follow directly from (2.1) by comparing coefficients after writing F(q)=A(q)F(q^3), adding this short derivation would make the paper self-contained. Likewise, Propositions 2.2 and 2.3 are asserted with 'it is not difficult to see' but no bijection is given; because Proposition 2.2 is the bridge from the generating function to the partition count used in Corollary 3.2, a sentence explaining the expansion of (2.1) as choices of marks would be appropriate.
[2, Corollary 2.5] In equation (2.7), the symbol N is undefined; it should be S(k−1; Z).
[4, Proposition 4.6] The statement 'Q(1)_{2n}(−2)=0 for all n ≥ 0' should read 'for all n ≥ 1', since Q(1)_0(z)=0 is the zero polynomial and not a meaningful zero statement.
[1, Examples 1–2] The overline and tilde markings are typographically indistinguishable in the running text (e.g., in Example 1 the last two partitions appear identical). Please use two clearly distinct diacritics throughout.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the perfect-number result is a genuine consequence of the generating function and recurrences; the only self-citation is to an independent prior theorem.

full rationale

The derivation chain is self-contained. The paper defines S(n;Z) by the generating function (2.1); Propositions 2.2 and 2.3 are direct readings of the product, not substantive inputs. The recurrences (2.4)-(2.6) are cited from the authors' earlier [1, Thm 4.3], but that is a parameter-free general theorem about restricted multicolor b-ary partitions, and the recurrences are immediate consequences of (2.1) by comparing coefficients in F(q)=A(q)F(q^3) with A(q)=(1+wq)(1+xq)(1+yq+zq^2). Thus the citation is replaceable by a one-line derivation and does not assume the perfect-number target. Proposition 3.1 supplies a complete induction from (2.4)-(2.5) to the second-order recurrences; specializing to Z0=(1,1,1,1) gives q_n=6q_{n-1}-8q_{n-2}, whose characteristic roots 2 and 4 directly yield q_n=2^{n-1}(2^n-1). Corollary 3.2 then invokes the Euclid-Euler theorem, an external classical result, to identify these values as even perfect numbers for Mersenne-prime exponents. Nothing in the recurrences, the W1,W2 coefficients, or the induction is fitted to the perfect-number subsequence; the perfect numbers enter only after the closed form is derived. The informal 'it is not difficult to see' identifications are unproved but are definitional transcriptions of the coloring conditions, not circular reductions. Hence no circular step exists.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted numerical parameters: the closed forms are derived from recurrences rather than tuned, the variables w,x,y,z carry explicit combinatorial statistics, and the subsequence indices (3^n - 3)/2 and (3^n - 1)/2 are selected by the mod-3 structure of the recurrences, not by fitting. The imported content is the encoding of rule (1.1) into (2.1), the recurrence system from [1], and standard external theorems (Euclid-Euler, Chebyshev). No new entities are postulated: the colored partition model is a definition, and the polynomials are derived objects.

assumptions (4)

domain assumption The product (2.1) exactly encodes the partition rule (1.1), so the coefficient S(n;Z) is the number of restricted colored base-3 partitions (Propositions 2.2 and 2.3).
Stated as 'it is not difficult to see'; no explicit bijection is given, and the four-variable weighting (overline, tilde, single unmarked, pair of unmarked) must match the three-color restriction including the cap of two unmarked copies.
domain assumption The recurrences (2.4)-(2.6) of Proposition 2.4, imported from the authors' own [1, Theorem 4.3].
These three recurrences are the engine for all later results and are not re-derived here; they are published peer-reviewed results, so this is cumulative reliance rather than circularity, but it is load-bearing and self-cited.
standard math Euclid-Euler theorem: every even perfect number equals 2^(p-1)(2^p - 1) with 2^p - 1 prime.
Used in Corollary 3.2 to convert the closed form q_n = 2^(n-1)(2^n - 1) into the perfect-number statement and its converse.
standard math Standard Chebyshev facts: real zeros in (-1,1) for U_n and T_n, generating functions (3.13), the Pell-type identity, divisibility of U_n, and explicit zero formulas (6.5).
These underpin Propositions 4.6, 5.3, 5.6 and Corollary 6.3; sourced to the textbook [6].

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Cite this review

Pith. "Pith review of Colored base-3 partitions, sequences of polynomials, and perfect numbers." pith.science (2026). https://pith.science/paper/DED7RYI2

@misc{pith2026250903147,
  author       = {Pith},
  title        = {Pith review of: Colored base-3 partitions, sequences of polynomials, and perfect numbers},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DED7RYI2}},
  note         = {Machine review of arXiv:2509.03147}
}

read the original abstract

Motivated by the observation that the counting function of a certain base-3 colored partition contains the even perfect numbers as a subsequence, we begin by defining a sequence of polynomials in four variables and discuss their properties and combinatorial interpretations. We then concentrate on certain subsequences that are related to the Chebyshev polynomials of both kinds. Finally, we consider several sequences of single-variable polynomials that have meaningful combinatorial interpretations as well as interesting zero distributions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 7 canonical work pages

[1]

Dilcher and L

K. Dilcher and L. Ericksen, Polynomial analogues of restricted multicolor b-ary partition functions, Int. J. Number Theory 17 (2021), no. 2, 371–391

work page 2021
[2]

Dilcher and L

K. Dilcher and L. Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, Ramanujan J. 68, 44 (2025). https://doi.org/10.1007/s11139-025-01187-3

work page doi:10.1007/s11139-025-01187-3 2025
[3]

W. J. Keith, Restricted k-color partitions, Ramanujan J. 40 (2016), 71–92

work page 2016
[4]

Laradji and A

A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, J. Algebra 278 (2004), no. 1, 342–359

work page 2004
[5]

(2011), The On-Line Encyclopedia of Integer Sequences , http://oeis.org

OEIS Foundation Inc. (2011), The On-Line Encyclopedia of Integer Sequences , http://oeis.org

work page 2011
[6]

T. J. Rivlin, Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory. Second Edition. Wiley, New York, 1990

work page 1990
[7]

Ulas and B

M. Ulas and B. ˙Zmija, On p-adic valuations of colored p-ary partitions, Monatsh. Math. 188 (2019), no. 2, 351–368. Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Sco- tia, B3H 4R2, Canada Email address : dilcher@mathstat.dal.ca 1212 Forest Drive, Millville, NJ 08332-2512, USA Email address : LE22@cornell.edu

work page 2019

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

[1] [1]

Dilcher and L

K. Dilcher and L. Ericksen, Polynomial analogues of restricted multicolor b-ary partition functions, Int. J. Number Theory 17 (2021), no. 2, 371–391

work page 2021

[2] [2]

Dilcher and L

K. Dilcher and L. Ericksen, Polynomials and algebraic curves related to certain binary and b-ary overpartitions, Ramanujan J. 68, 44 (2025). https://doi.org/10.1007/s11139-025-01187-3

work page doi:10.1007/s11139-025-01187-3 2025

[3] [3]

W. J. Keith, Restricted k-color partitions, Ramanujan J. 40 (2016), 71–92

work page 2016

[4] [4]

Laradji and A

A. Laradji and A. Umar, Combinatorial results for semigroups of order-preserving partial transformations, J. Algebra 278 (2004), no. 1, 342–359

work page 2004

[5] [5]

(2011), The On-Line Encyclopedia of Integer Sequences , http://oeis.org

OEIS Foundation Inc. (2011), The On-Line Encyclopedia of Integer Sequences , http://oeis.org

work page 2011

[6] [6]

T. J. Rivlin, Chebyshev Polynomials. From Approximation Theory to Algebra and Number Theory. Second Edition. Wiley, New York, 1990

work page 1990

[7] [7]

Ulas and B

M. Ulas and B. ˙Zmija, On p-adic valuations of colored p-ary partitions, Monatsh. Math. 188 (2019), no. 2, 351–368. Department of Mathematics and Statistics, Dalhousie University, Halifax, Nova Sco- tia, B3H 4R2, Canada Email address : dilcher@mathstat.dal.ca 1212 Forest Drive, Millville, NJ 08332-2512, USA Email address : LE22@cornell.edu

work page 2019