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arxiv: 2604.05403 · v1 · submitted 2026-04-07 · 🧮 math.NT

Proof of a conjecture of Banerjee,Bringmann and Bachraoui on infinite families of congruences

Junjie Sun , Olivia X.M. Yao This is my paper

Pith reviewed 2026-05-10 19:11 UTC · model grok-4.3

classification 🧮 math.NT
keywords partition congruencestwo-color partitionsmock theta functionsmodular formsWatson's identityRamanujan congruenceslimiting sequence
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The pith

The conjecture on infinite families of congruences for the limiting sequence of restricted two-color partitions is proved.

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

This paper proves the conjecture of Banerjee, Bringmann and Bachraoui that a certain limiting sequence of restricted two-color partitions satisfies infinite families of congruences modulo 4 and 8. The proof combines the modular-form and mock theta function relations already established for the generating function with an identity due to Watson. The result directly implies the two earlier conjectures of Andrews and Bachraoui on Ramanujan-type congruences and a vanishing identity. A reader would care because these congruences expose arithmetic structure in partition functions that is not visible from the definition alone.

Core claim

Using the generating-function identities and modular-form relations established by Banerjee, Bringmann and Bachraoui together with Watson's identity, the paper shows that the limiting sequence satisfies the conjectured infinite families of congruences modulo 4 and 8.

What carries the argument

Watson's identity applied to the modular-form and mock theta expression for the generating function of the limiting sequence.

Load-bearing premise

The generating-function identities and modular-form relations established by Banerjee, Bringmann and Bachraoui are correct and Watson's identity applies directly to the limiting sequence without additional restrictions.

What would settle it

A direct computation showing that the partition count for some large n in one of the claimed arithmetic progressions fails to satisfy the stated congruence modulo 4 or 8.

read the original abstract

Recently, Andrews and Bachraoui investigated congruences for certain restricted two-color partitions. They made two conjectures for Ramanujan type congruences and a vanishing identity for the limiting sequence. Very recently, Banerjee, Bringmann and Bachraoui confirmed these three conjectures by relating the corresponding generating function to modular forms and mock theta functions. At the end of their paper, they posed a conjecture on infinite families of congruences modulo 4 and 8 for the limiting sequence. The Banerjee-Bringmann-Bachraoui's conjecture implies the two conjectures given by Andrews and Bachraoui. In this note, we settle Banerjee-Bringmann-Bachraoui's conjecture on infinite famlies of congruences based on Banerjee-Bringmann-Bachraoui's results and an identity due to Waston.

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

1 major / 2 minor

Summary. The manuscript claims to settle the conjecture of Banerjee, Bringmann and Bachraoui on infinite families of congruences modulo 4 and 8 for the limiting sequence of restricted two-color partitions. It asserts that the result follows directly from the generating-function identities and modular-form relations already established by Banerjee et al. together with an application of Watson's q-series identity.

Significance. If the central claim holds, the note would confirm the conjecture, which in turn implies the two earlier conjectures of Andrews and Bachraoui on Ramanujan-type congruences and a vanishing identity. This would add a modest but concrete increment to the literature on partition congruences involving mock theta functions.

major comments (1)
  1. [main argument (following the abstract)] The manuscript states that the target congruences follow from Banerjee-Bringmann-Bachraoui's generating-function identities combined with Watson's identity, but supplies no explicit substitution, limit interchange, or verification that the limiting sequence satisfies the parameter restrictions (e.g., non-vanishing denominators or radius-of-convergence conditions) required for Watson's identity to apply. This step is load-bearing for the central claim.
minor comments (2)
  1. [abstract] The abstract contains two typographical errors: 'familes' should read 'families' and 'Waston' should read 'Watson'.
  2. [main argument] The specific form of Watson's identity invoked (including any citation or equation number) should be stated explicitly so that the reader can check the parameter restrictions directly.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: The manuscript states that the target congruences follow from Banerjee-Bringmann-Bachraoui's generating-function identities combined with Watson's identity, but supplies no explicit substitution, limit interchange, or verification that the limiting sequence satisfies the parameter restrictions (e.g., non-vanishing denominators or radius-of-convergence conditions) required for Watson's identity to apply. This step is load-bearing for the central claim.

    Authors: We agree that the current exposition is too terse on this point. In the revised version we will insert an explicit paragraph that (i) records the precise substitution of the generating-function identities from Banerjee-Bringmann-Bachraoui into Watson's q-series identity, (ii) justifies the interchange of the limiting process with the series summation by appealing to the uniform convergence estimates already established in that paper, and (iii) verifies that the resulting parameters satisfy the non-vanishing-denominator and radius-of-convergence hypotheses of Watson's identity. revision: yes

Circularity Check

0 steps flagged

No circularity; proof invokes independent external results and identity

full rationale

The paper's derivation explicitly rests on generating-function identities and modular-form relations from Banerjee-Bringmann-Bachraoui (distinct authors) together with Watson's q-series identity. No self-citations appear, no parameters are fitted inside the paper and then renamed as predictions, and no step defines the target congruences in terms of themselves or reduces them by construction to the paper's own inputs. The argument is therefore a standard external-proof structure whose load-bearing steps remain independent of the present manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the correctness of the generating-function and modular-form relations proved by Banerjee, Bringmann and Bachraoui together with the applicability of Watson's identity; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption The generating-function identities and modular-form relations established by Banerjee, Bringmann and Bachraoui hold for the relevant restricted partitions.
    Invoked in the abstract as the foundation for the new proof.
  • domain assumption Watson's identity applies directly to the limiting sequence without further restrictions.
    Cited as the second ingredient that completes the proof.

pith-pipeline@v0.9.0 · 5444 in / 1280 out tokens · 52375 ms · 2026-05-10T19:11:59.698147+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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