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arxiv: 2510.19242 · v2 · submitted 2025-10-22 · 🧮 math.CO · math.NT

Congruences modulo powers of 3 for generalized Frobenius partitions CPsi_(6,0)

Dandan Chen , Siyu Yin This is my paper

Pith reviewed 2026-05-18 05:21 UTC · model grok-4.3

classification 🧮 math.CO math.NT
keywords generalized Frobenius partitionscongruences modulo powers of 3Atkin-Lehner involutioncolored partitionspartition generating functionsmodular forms
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The pith

The (6,0)-colored Frobenius partition function cψ_{6,0}(n) satisfies congruences modulo powers of 3.

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

The paper links the generating functions for cψ_{6,3}(n) and cψ_{6,0}(n) through an Atkin-Lehner involution. This link transfers congruence results previously shown for related 6-colored partition functions to the (6,0) case. A reader cares because these arithmetic patterns in partition counts often reflect hidden modular structures that organize counting problems across number theory. The work builds directly on earlier congruences for cφ_6(n) by applying the same modular machinery to a new family of partitions.

Core claim

By applying an Atkin-Lehner involution to the generating function of cψ_{6,3}(n), the authors obtain the generating function for cψ_{6,0}(n) and thereby prove that cψ_{6,0}(n) obeys congruences modulo powers of 3.

What carries the argument

The Atkin-Lehner involution that transforms the generating function of cψ_{6,3}(n) into the generating function of cψ_{6,0}(n) while preserving the modular properties needed for congruence transfer.

If this is right

  • cψ_{6,0}(n) obeys the same family of congruences modulo 3^k as the related function cφ_6(n).
  • The arithmetic regularity allows prediction of cψ_{6,0}(n) in certain residue classes without direct computation.
  • The connection extends the earlier congruence results to a new pair of colored Frobenius partition functions.

Where Pith is reading between the lines

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

  • Similar involution arguments could relate other pairs of colored partition functions and yield congruences for additional cases.
  • The technique might extend to congruences modulo primes other than 3 if the modular forms behave analogously.
  • Explicit formulas or recurrence relations for cψ_{6,0}(n) could be derived from the transferred generating function.

Load-bearing premise

The Atkin-Lehner involution maps the generating function of cψ_{6,3}(n) to that of cψ_{6,0}(n) in a way that preserves the modular properties required to carry over the known congruences.

What would settle it

A specific integer n where the value of cψ_{6,0}(n) fails to satisfy one of the stated congruences modulo 9 or 27.

read the original abstract

In 1984, Andrews introduced the family of partition functions \(c\phi_k(n)\), which counts the number of generalized Frobenius partitions of \(n\) with \(k\) colors. In previous work, we proved a conjecture on congruences for \(c\phi_6(n)\) modulo powers of 3. In this paper, we consider the \((6,0)\)-colored Frobenius partition functions \(c\psi_{6,0}(n)\). We establish a connection between the generating functions of \(c\psi_{6,3}(n)\) and \(c\psi_{6,0}(n)\) via an Atkin-Lehner involution, and prove congruences modulo powers of 3 for \(c\psi_{6,0}(n)\).

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.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to prove congruences modulo powers of 3 for the generalized Frobenius partition function cψ_{6,0}(n) by establishing a connection between the generating functions of cψ_{6,3}(n) and cψ_{6,0}(n) via an Atkin-Lehner involution, allowing transfer of congruence results from the authors' prior work on cφ_6(n).

Significance. If the Atkin-Lehner involution preserves the necessary modular-form properties (weight, level, and nebentypus), the result would extend the arithmetic theory of colored generalized Frobenius partitions and demonstrate a useful technique for relating variants of these functions through involutions on their generating functions.

major comments (2)
  1. [§3] §3 (connection via Atkin-Lehner): the claim that the involution maps the generating function of cψ_{6,3}(n) to that of cψ_{6,0}(n) while preserving the precise level and character required for the 3-adic congruence lifting from the cφ_6 work is not accompanied by an explicit verification of the transformed form's nebentypus or level; if these change, the Hecke operators or filtration used previously may not apply directly.
  2. [Theorem 5.1] Theorem 5.1 (main congruence statement): the congruences for cψ_{6,0}(n) mod 3^k are derived by direct transfer; without a concrete check that the image lies in a space compatible with the prior machinery (or an adjustment of the filtration), the argument risks circularity with respect to the external properties invoked.
minor comments (2)
  1. [Introduction] The introduction would benefit from a brief explicit example computing cψ_{6,0}(n) for small n to illustrate the definition.
  2. Notation for the colored partitions and the precise statement of the Atkin-Lehner operator could be made more uniform across sections.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points where additional explicit verification would strengthen the presentation. We have revised the paper to address both major comments by adding the requested computations and checks. Our responses to the individual comments follow.

read point-by-point responses

  1. Referee: [§3] §3 (connection via Atkin-Lehner): the claim that the involution maps the generating function of cψ_{6,3}(n) to that of cψ_{6,0}(n) while preserving the precise level and character required for the 3-adic congruence lifting from the cφ_6 work is not accompanied by an explicit verification of the transformed form's nebentypus or level; if these change, the Hecke operators or filtration used previously may not apply directly.

    Authors: We agree that an explicit verification strengthens the argument. In the revised Section 3 we now include a direct computation of the image under the Atkin-Lehner involution: we apply the operator to the q-expansion of the generating function for cψ_{6,3}(n), compute the resulting level and nebentypus explicitly, and confirm that both remain identical to those of the modular form treated in our earlier work on cφ_6. Consequently the same Hecke operators and 3-adic filtration apply without adjustment. revision: yes

  2. Referee: [Theorem 5.1] Theorem 5.1 (main congruence statement): the congruences for cψ_{6,0}(n) mod 3^k are derived by direct transfer; without a concrete check that the image lies in a space compatible with the prior machinery (or an adjustment of the filtration), the argument risks circularity with respect to the external properties invoked.

    Authors: We appreciate the referee's caution regarding possible circularity. The revised proof of Theorem 5.1 now contains an explicit lemma verifying that the Atkin-Lehner image lies in the precise space of modular forms (same weight, level, and character) for which the congruence machinery of the cφ_6 paper was established. Because this compatibility is checked directly on the image rather than assumed from the original form, the transfer of the 3-adic congruences is justified without circular reasoning. No adjustment to the filtration is required. revision: yes

Circularity Check

1 steps flagged

Central congruences for cψ_{6,0}(n) transferred from authors' prior self-work via Atkin-Lehner connection

specific steps
  1. self citation load bearing [Abstract]
    "In previous work, we proved a conjecture on congruences for cφ_6(n) modulo powers of 3. In this paper, we consider the (6,0)-colored Frobenius partition functions cψ_{6,0}(n). We establish a connection between the generating functions of cψ_{6,3}(n) and cψ_{6,0}(n) via an Atkin-Lehner involution, and prove congruences modulo powers of 3 for cψ_{6,0}(n)."

    The congruences claimed for cψ_{6,0}(n) are proved by invoking the Atkin-Lehner map to transfer modular properties and results directly from the authors' prior paper on cφ_6(n) (and implicitly cψ_{6,3}). This makes the self-citation load-bearing, as the central theorem reduces to assuming the prior self-result applies verbatim once the connection is stated, without independent re-derivation of the 3-power congruences in the present work.

full rationale

The paper establishes a new connection between generating functions of cψ_{6,3}(n) and cψ_{6,0}(n) using an Atkin-Lehner involution, which constitutes independent content. However, the congruences modulo powers of 3 are obtained by transferring results from the authors' own previous work on cφ_6(n), making the self-citation load-bearing for the main result. The derivation chain therefore contains some dependence on prior self-authored results rather than being fully self-contained within this manuscript, though the connection step itself does not reduce by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard background results from partition theory and modular forms. No free parameters are introduced. No new entities are postulated.

axioms (2)
  • domain assumption The generating functions for cψ_{6,3}(n) and cψ_{6,0}(n) admit an Atkin-Lehner involution that relates them algebraically.
    Invoked to establish the connection between the two functions.
  • standard math Standard properties of modular forms and generating-function identities hold for these colored partition functions.
    Used throughout the proof of the congruences.

pith-pipeline@v0.9.0 · 5666 in / 1449 out tokens · 37241 ms · 2026-05-18T05:21:58.295479+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 2 internal anchors

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

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