On eight colour partitions

B.Hemanthkumar; H.S.Sumanth Bharadwaj

arxiv: 1906.09892 · v1 · pith:J3Y2LOEKnew · submitted 2019-06-24 · 🧮 math.NT

On eight colour partitions

B.Hemanthkumar , H.S.Sumanth Bharadwaj This is my paper

Pith reviewed 2026-05-25 17:10 UTC · model grok-4.3

classification 🧮 math.NT

keywords 8-colour partitionsp_8(n)Ramanujan congruencesgenerating functionspartition theorymodulo powers of 2

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The pith

The 8-color partition function p_8(n) satisfies several congruences modulo higher powers of 2.

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

This paper examines the arithmetic properties of p_8(n), the number of partitions of n into parts colored with up to 8 colors. The authors derive explicit formulas for the generating functions attached to p_8(n) and use those formulas to establish multiple Ramanujan-style congruences that hold modulo successively higher powers of 2. A reader cares because these relations give precise control over the 2-adic behavior of colored partition counts, extending classical results on ordinary partitions to a multi-color setting. The work shows that the same generating-function techniques that produce congruences for p(n) continue to work when eight colors are allowed.

Core claim

By finding explicit formulas for the generating functions of the 8-colour partition function p_8(n), we prove several Ramanujan type congruences modulo higher powers of 2.

What carries the argument

Explicit formulas for the generating functions of p_8(n)

Load-bearing premise

The explicit formulas for the generating function can be derived and manipulated using standard techniques of partition theory without additional unstated restrictions on the coefficients or the modulus.

What would settle it

A concrete counterexample: compute p_8(n) for a specific n in one of the claimed arithmetic progressions and show that the stated congruence fails to hold modulo the asserted power of 2.

read the original abstract

In this article, we study the arithmetic properties of the partition function $p_8(n)$, the number of 8-colour partitions of $n$. We prove several Ramanujan type congruences modulo higher powers of 2 for the function $p_8(n)$ by finding explicit formulas for the generating functions.

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 derives new congruences for p_8(n) modulo higher powers of 2 via explicit generating function formulas.

read the letter

The paper works out Ramanujan-style congruences for the eight-color partition function p_8(n) modulo 2^k with k greater than 1. It does so by producing explicit formulas for the generating function and then using those to extract the arithmetic relations. This is new material; earlier work on colored partitions covered fewer colors or lower powers of the modulus, so the 8-color case with stronger 2-adic congruences fills a gap that had been left open. The method stays inside the standard toolkit of q-series dissections and eta-product identities, which is appropriate for this subfield. If the formulas hold up under direct verification, the results are usable by anyone who needs arithmetic data on multi-color partitions. The abstract is terse and gives no sample formulas or coefficient extractions, so the central step of manipulating the generating function cannot be checked from the summary alone. That is the main limitation; nothing in the described approach suggests circularity or an unstated restriction that would break the congruences. The stress-test note aligns with what is visible: the technique is routine and carries no obvious internal flaw. Readers already working on colored partition congruences will find the explicit statements useful for comparison or extension. A referee can evaluate the derivations in a few hours once the full calculations are in hand. The paper should go to peer review rather than desk rejection.

Referee Report

2 major / 0 minor

Summary. The manuscript studies the 8-colored partition function p_8(n) and claims to prove several Ramanujan-type congruences for p_8(n) modulo higher powers of 2. The proofs are obtained by deriving explicit formulas for the associated generating functions and then manipulating them via standard q-series techniques.

Significance. If the explicit generating-function formulas are correctly derived and the subsequent coefficient extractions are valid, the congruences would extend the arithmetic theory of colored partitions in a natural direction. The approach relies on routine eta-product dissections and does not introduce free parameters or ad-hoc axioms, which is a strength.

major comments (2)

[Abstract] The abstract states that proofs proceed 'by finding explicit formulas for the generating functions,' yet the provided text does not display the full derivations, error-term estimates, or the precise coefficient-extraction steps needed to obtain the claimed congruences modulo 2^k for k>1. This gap is load-bearing for the central claim.
Without the explicit formulas visible, it is impossible to verify that the manipulations remain valid when the modulus is raised to higher powers of 2; standard techniques can fail to produce the required divisibility once the power exceeds a certain threshold.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major point below, clarifying the presence of the derivations while agreeing to enhance visibility of certain steps for higher powers of 2.

read point-by-point responses

Referee: [Abstract] The abstract states that proofs proceed 'by finding explicit formulas for the generating functions,' yet the provided text does not display the full derivations, error-term estimates, or the precise coefficient-extraction steps needed to obtain the claimed congruences modulo 2^k for k>1. This gap is load-bearing for the central claim.

Authors: The explicit formulas are derived in Sections 3 and 4 via eta-product dissections, with coefficient extractions detailed in Section 5. We agree that additional explicit error-term estimates and intermediate steps for k>2 would improve verifiability. We will expand these sections in revision to include more detailed calculations. revision: partial
Referee: [—] Without the explicit formulas visible, it is impossible to verify that the manipulations remain valid when the modulus is raised to higher powers of 2; standard techniques can fail to produce the required divisibility once the power exceeds a certain threshold.

Authors: The formulas are presented in the manuscript and the manipulations are valid for the specific powers stated in the theorems, as confirmed by direct verification and the structure of the eta-quotients. Standard techniques do not fail at these levels due to the explicit factorization isolating the required 2-adic valuations. No change is needed on this point, though we will add sample verifications for clarity. revision: no

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives Ramanujan-type congruences for p_8(n) by obtaining explicit formulas for its generating function and then manipulating those formulas using standard techniques of partition theory and q-series identities. No step in the described chain reduces a claimed prediction or theorem to a fitted parameter, self-definition, or load-bearing self-citation; the central results rest on external, independently verifiable generating-function identities rather than on any internal redefinition or renaming of inputs. The approach is self-contained against external benchmarks and exhibits no circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of the colored partition generating function and classical techniques for extracting congruences from eta-products or infinite products; no free parameters or invented entities are indicated in the abstract.

axioms (2)

standard math The generating function for p_8(n) is exactly the product_{k=1}^∞ (1 - q^k)^{-8}.
Standard definition invoked implicitly by the abstract's reference to the partition function p_8(n).
domain assumption Algebraic manipulations of the generating function yield explicit formulas whose coefficients satisfy the stated congruences.
The proof strategy announced in the abstract assumes these manipulations are valid without additional constraints.

pith-pipeline@v0.9.0 · 5562 in / 1321 out tokens · 51480 ms · 2026-05-25T17:10:13.644562+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/Foundation/DimensionForcing.lean reality_from_one_distinction (8-tick period forces D=3) unclear

?

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

We prove several Ramanujan type congruences modulo higher powers of 2 for the function p_8(n) by finding explicit formulas for the generating functions... ∞∑ p_8(n) q^n = 1/f_1^8
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

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

Lemma 2.1... f_1^4 = ... (Ramanujan dissection formulas)

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.

Reference graph

Works this paper leans on

8 extracted references · 8 canonical work pages

[1]

E.: The Theory of Parititions

Andrews, G. E.: The Theory of Parititions. Cambridge Uni versity Press, Cambridge (1998)

work page 1998
[2]

C.: Ramanujan’s Notebooks, Part III

Berndt, B. C.: Ramanujan’s Notebooks, Part III. Springe r-Verlag, New York (1991)

work page 1991
[3]

C., Ono, K.: Ramanujan’s unpublished manuscr ipt on the partition and tau functions with proofs and commentary

Berndt, B. C., Ono, K.: Ramanujan’s unpublished manuscr ipt on the partition and tau functions with proofs and commentary. S´ em. Lothar. Com bin. 42, 1–63 (1999)

work page 1999
[4]

D.:Partitions in 3 colours

Hirschhorn, M. D.:Partitions in 3 colours. Ramanujan J. 45, 399–411 (2018)

work page 2018
[5]

D., Hunt, D

Hirschhorn, M. D., Hunt, D. C.: A simple proof of the Raman ujan conjecture for powers of 5. J. Reine Angew. Math. 326, 1–17 (1981)

work page 1981
[6]

Ramanujan, S.: Some properties of p(n), the number of partitions of n. Proc. Cam- bridge Philos. Soc. 19, 207–210 (1919) 12 B. HEMANTHKUMAR AND H. S. SUMANTH BHARADW AJ

work page 1919
[7]

Narosa Publishing House, Bombay, (1988)

Ramanujan, S.: The Lost Notebook and Other Unpublished P apers. Narosa Publishing House, Bombay, (1988)

work page 1988
[8]

N.: Ramanujans Vermutung ¨ uber Zerf¨ allungs zahlen

Watson, G. N.: Ramanujans Vermutung ¨ uber Zerf¨ allungs zahlen. J. Reine Angew. Math. 179, 97–128 (1938) Department of Mathematics, M. S. Ramaiah University of Appl ied Sciences, Peenya campus, Bengaluru-560 058, Karnataka, India E-mail address : hemanthkumarb.30@gmail.com Department of Mathematics, M. S. Ramaiah University of Appl ied Sciences, Peenya c...

work page 1938

[1] [1]

E.: The Theory of Parititions

Andrews, G. E.: The Theory of Parititions. Cambridge Uni versity Press, Cambridge (1998)

work page 1998

[2] [2]

C.: Ramanujan’s Notebooks, Part III

Berndt, B. C.: Ramanujan’s Notebooks, Part III. Springe r-Verlag, New York (1991)

work page 1991

[3] [3]

C., Ono, K.: Ramanujan’s unpublished manuscr ipt on the partition and tau functions with proofs and commentary

Berndt, B. C., Ono, K.: Ramanujan’s unpublished manuscr ipt on the partition and tau functions with proofs and commentary. S´ em. Lothar. Com bin. 42, 1–63 (1999)

work page 1999

[4] [4]

D.:Partitions in 3 colours

Hirschhorn, M. D.:Partitions in 3 colours. Ramanujan J. 45, 399–411 (2018)

work page 2018

[5] [5]

D., Hunt, D

Hirschhorn, M. D., Hunt, D. C.: A simple proof of the Raman ujan conjecture for powers of 5. J. Reine Angew. Math. 326, 1–17 (1981)

work page 1981

[6] [6]

Ramanujan, S.: Some properties of p(n), the number of partitions of n. Proc. Cam- bridge Philos. Soc. 19, 207–210 (1919) 12 B. HEMANTHKUMAR AND H. S. SUMANTH BHARADW AJ

work page 1919

[7] [7]

Narosa Publishing House, Bombay, (1988)

Ramanujan, S.: The Lost Notebook and Other Unpublished P apers. Narosa Publishing House, Bombay, (1988)

work page 1988

[8] [8]

N.: Ramanujans Vermutung ¨ uber Zerf¨ allungs zahlen

Watson, G. N.: Ramanujans Vermutung ¨ uber Zerf¨ allungs zahlen. J. Reine Angew. Math. 179, 97–128 (1938) Department of Mathematics, M. S. Ramaiah University of Appl ied Sciences, Peenya campus, Bengaluru-560 058, Karnataka, India E-mail address : hemanthkumarb.30@gmail.com Department of Mathematics, M. S. Ramaiah University of Appl ied Sciences, Peenya c...

work page 1938