Congruences modulo powers of $3$ for $6$-colored generalized Frobenius partitions

Dandan Chen; Siyu Yin

arxiv: 2504.04983 · v2 · submitted 2025-04-07 · 🧮 math.CO · math.NT

Congruences modulo powers of 3 for 6-colored generalized Frobenius partitions

Dandan Chen , Siyu Yin This is my paper

Pith reviewed 2026-05-22 20:47 UTC · model grok-4.3

classification 🧮 math.CO math.NT

keywords generalized Frobenius partitionscolored partitionspartition congruencesmodulo powers of 3generating functions

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The pith

The six-colored generalized Frobenius partition function satisfies congruences modulo powers of 3 as stated in a revised conjecture.

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

The paper proves a collection of arithmetic relations for the function that counts generalized Frobenius partitions using six colors. It settles a revised form of an earlier conjecture by showing that the function values are congruent to zero or to other specific residues when reduced modulo successive powers of three, for n in certain residue classes. The proof proceeds by direct work with the generating function to isolate coefficients that obey the relations. A reader would care because these relations reveal hidden divisibility patterns that persist across different color counts in the same family of partition functions.

Core claim

The function cφ_6(n) obeys the revised set of congruences modulo 3^k, established through manipulation of its generating function that isolates the coefficients satisfying the required divisibility conditions.

What carries the argument

The ordinary generating function for the six-colored generalized Frobenius partitions, together with its decomposition that extracts coefficients divisible by the required power of 3.

Load-bearing premise

The same technique used for related partition functions carries over directly to the six-color generating function and produces the stated congruences without extra adjustments.

What would settle it

An integer n in one of the residue classes for which cφ_6(n) fails to satisfy one of the claimed congruences modulo 3^k.

read the original abstract

In $1984$, Andrews introduced the family of partition functions $c\phi_k(n)$, which enumerate generalized Frobenius partitions of $n$ with $k$ colors. In $2016$, Gu, Wang, and Xia established several congruences for $c\phi_6(n)$ and proposed a conjecture concerning congruences modulo powers of $3$ for this function. In this paper, we resolve a revised version of their conjecture by employing an approach analogous to that developed by Banerjee and Smoot.

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

Chen and Yin prove a revised version of the Gu-Wang-Xia conjecture on cφ6(n) mod 3^k by adapting the Banerjee-Smoot dissection method.

read the letter

Hi, the main point is that this paper turns a revised form of the 2016 Gu-Wang-Xia conjecture into theorems by applying an analogous sequence of 3-dissection operators and 3-adic estimates to the six-color generating function. That is the actual advance: new proofs where only conjectures existed before. The adaptation itself looks reasonable on the surface because the six-color case shares the same basic structure as the functions Banerjee and Smoot handled, so the same operator steps can be carried over without inventing new machinery. Credit is due for carrying the method through to explicit congruences rather than stopping at the abstract level. The soft spot is the revision of the original conjecture. The paper states it works on a revised version, but the letter should make clear exactly which statements were altered and why; if the changes exclude cases that were part of the original conjecture, the result is narrower than it first appears. The stress-test worry about extra identities or exclusions does not seem to bite here once the generating function is written out, because the key recurrence relations after the first dissection match the source technique closely enough. No circularity or free parameters show up in the argument. This paper is for people who follow colored partition congruences and want the next concrete case filled in. It is narrow but cleanly executed, so a serious referee should see it. I would send it to review with the request that the authors spell out the precise differences between the original and revised conjectures.

Referee Report

1 major / 1 minor

Summary. The manuscript resolves a revised version of the Gu-Wang-Xia conjecture from 2016 on congruences modulo powers of 3 for the 6-colored generalized Frobenius partition function cφ6(n), by applying an approach analogous to the one developed by Banerjee and Smoot.

Significance. If the central derivations hold, the work would complete the resolution of the revised conjecture, thereby extending the arithmetic theory of colored generalized Frobenius partitions and confirming that the Banerjee-Smoot dissection-and-estimate technique transfers to the k=6 generating function without alteration of the stated congruences.

major comments (1)

[Introduction] Introduction: the manuscript refers to resolving a 'revised version' of the Gu-Wang-Xia conjecture but does not state the original conjecture or the precise nature of the revision. This information is load-bearing for assessing the scope of the result.

minor comments (1)

The abstract and introduction should include a brief explicit statement of the key generating-function identity or recurrence that is transferred from the Banerjee-Smoot setting to the six-color case.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for their recommendation of minor revision. We address the major comment below.

read point-by-point responses

Referee: [Introduction] Introduction: the manuscript refers to resolving a 'revised version' of the Gu-Wang-Xia conjecture but does not state the original conjecture or the precise nature of the revision. This information is load-bearing for assessing the scope of the result.

Authors: We agree with the referee that providing the original statement of the Gu-Wang-Xia conjecture and clarifying the precise revision would enhance the reader's understanding of the scope of our result. In the revised version of the manuscript, we will add this information to the introduction. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external analogy

full rationale

The paper states it resolves a revised conjecture for cφ6(n) mod 3^k by employing an approach analogous to Banerjee and Smoot (distinct authors). No quoted equations or claims reduce the central result to a self-definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The derivation chain is presented as transferring an external technique, making the result self-contained against the cited prior work rather than internally forced.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are visible.

pith-pipeline@v0.9.0 · 5609 in / 887 out tokens · 38302 ms · 2026-05-22T20:47:38.061192+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We resolve a revised version of their conjecture by employing an approach analogous to that developed by Banerjee and Smoot... L2α−1 := UA(L2α−2) and L2α := UB(L2α−1)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

X^3 + a2(t)X^2 + a1(t)X + a0(t) = (X−t0)(X−t1)(X−t2)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Congruences modulo powers of $3$ for generalized Frobenius partitions $C\Psi_{6,0}$
math.CO 2025-10 unverdicted novelty 5.0

Proves congruences modulo powers of 3 for cψ_{6,0}(n) by connecting its generating function to cψ_{6,3}(n) via an Atkin-Lehner involution.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages · cited by 1 Pith paper · 1 internal anchor

[1]

G. E. Andrews,The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976; MR0557013

work page 1976
[2]

G. E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc.49(1984), no. 301, iv+44 pp.; MR0743546

work page 1984
[3]

A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J.8(1967), 14–32; MR0205958

work page 1967
[4]

A. O. L. Atkin and J. Lehner, Hecke operators on Γ 0(m), Math. Ann.185(1970), 134–160; MR0268123

work page 1970
[5]

Banerjee and N

K. Banerjee and N. A. Smoot, The localization method applied tok-elongated plane partitions and divisibility by 5, Math. Z.309(2025), no. 3, Paper No. 46, 51 pp.; MR4852261

work page 2025
[6]

N. D. Baruah and B. K. Sarmah, Generalized Frobenius partitions with 6 colors, Ramanujan J.38 (2015), no. 2, 361–382; MR3414497

work page 2015
[7]

F. G. Garvan, A simple proof of Watson’s partition congruences for powers of 7, J. Austral. Math. Soc. Ser. A36(1984), no. 3, 316–334; MR0733905

work page 1984
[8]

F. G. Garvan, A tutorial for the MAPLE ETA package, arXiv:1907.09130

work page internal anchor Pith review Pith/arXiv arXiv 1907
[9]

C. Gu, L. Wang and E. X. W. Xia, Congruences modulo powers of 3 for generalized Frobenius paritions with six colors, Acta Arith.175(2016), no. 3, 291–300; MR3557126

work page 2016
[10]

M. D. Hirschhorn and D. C. Hunt, A simple proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math.326(1981), 1–17; MR0622342

work page 1981
[11]

M. D. Hirschhorn, Some congruences for 6-colored generalized Frobenius partitions, Ramanujan J.40 (2016), no. 3, 463–471; MR3522077

work page 2016
[12]

Ramanujan, Some properties ofp(n), the number of partitions ofn, Proc

S. Ramanujan, Some properties ofp(n), the number of partitions ofn, Proc. Camb. Philos. Soc. 19, 214-216 (1919). MR2280868

work page 1919
[13]

D. Z. Tang, Congruence properties modulo powers of 3 for 6-colored generalized Frobeniuspartitions. Contrib. Discrete Math. 20 (2025), no. 1, 60–73. MR4900298

work page 2025
[14]

E. X. W. Xia, Proof of a conjecture of Baruah and Sarmah on generalized Frobenius partitions with 6 colors, J. Number Theory147(2015), 852–860; MR3276358

work page 2015
[15]

G. N. Watson, Ramanujans Vermutung ¨ uber Zerf¨ allungszahlen, J. Reine Angew. Math.179(1938), 97–128; MR1581588 Department of Mathematics, Shanghai University, People’s Republic of China Newtouch Center for Mathematics of Shanghai University, Shanghai, People’s Republic of China Email address:mathcdd@shu.edu.cn Department of Mathematics, East China Norma...

work page 1938

[1] [1]

G. E. Andrews,The theory of partitions, Encyclopedia of Mathematics and its Applications, Vol. 2, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976; MR0557013

work page 1976

[2] [2]

G. E. Andrews, Generalized Frobenius partitions, Mem. Amer. Math. Soc.49(1984), no. 301, iv+44 pp.; MR0743546

work page 1984

[3] [3]

A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J.8(1967), 14–32; MR0205958

work page 1967

[4] [4]

A. O. L. Atkin and J. Lehner, Hecke operators on Γ 0(m), Math. Ann.185(1970), 134–160; MR0268123

work page 1970

[5] [5]

Banerjee and N

K. Banerjee and N. A. Smoot, The localization method applied tok-elongated plane partitions and divisibility by 5, Math. Z.309(2025), no. 3, Paper No. 46, 51 pp.; MR4852261

work page 2025

[6] [6]

N. D. Baruah and B. K. Sarmah, Generalized Frobenius partitions with 6 colors, Ramanujan J.38 (2015), no. 2, 361–382; MR3414497

work page 2015

[7] [7]

F. G. Garvan, A simple proof of Watson’s partition congruences for powers of 7, J. Austral. Math. Soc. Ser. A36(1984), no. 3, 316–334; MR0733905

work page 1984

[8] [8]

F. G. Garvan, A tutorial for the MAPLE ETA package, arXiv:1907.09130

work page internal anchor Pith review Pith/arXiv arXiv 1907

[9] [9]

C. Gu, L. Wang and E. X. W. Xia, Congruences modulo powers of 3 for generalized Frobenius paritions with six colors, Acta Arith.175(2016), no. 3, 291–300; MR3557126

work page 2016

[10] [10]

M. D. Hirschhorn and D. C. Hunt, A simple proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math.326(1981), 1–17; MR0622342

work page 1981

[11] [11]

M. D. Hirschhorn, Some congruences for 6-colored generalized Frobenius partitions, Ramanujan J.40 (2016), no. 3, 463–471; MR3522077

work page 2016

[12] [12]

Ramanujan, Some properties ofp(n), the number of partitions ofn, Proc

S. Ramanujan, Some properties ofp(n), the number of partitions ofn, Proc. Camb. Philos. Soc. 19, 214-216 (1919). MR2280868

work page 1919

[13] [13]

D. Z. Tang, Congruence properties modulo powers of 3 for 6-colored generalized Frobeniuspartitions. Contrib. Discrete Math. 20 (2025), no. 1, 60–73. MR4900298

work page 2025

[14] [14]

E. X. W. Xia, Proof of a conjecture of Baruah and Sarmah on generalized Frobenius partitions with 6 colors, J. Number Theory147(2015), 852–860; MR3276358

work page 2015

[15] [15]

G. N. Watson, Ramanujans Vermutung ¨ uber Zerf¨ allungszahlen, J. Reine Angew. Math.179(1938), 97–128; MR1581588 Department of Mathematics, Shanghai University, People’s Republic of China Newtouch Center for Mathematics of Shanghai University, Shanghai, People’s Republic of China Email address:mathcdd@shu.edu.cn Department of Mathematics, East China Norma...

work page 1938