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New Congruences on Biregular Overpartitions

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims infinitely many mod-3 and mod-2^k congruence families for biregular overpartitions, proved by reducing generating functions to Hecke eigenforms.

desk verdict Theorem 8.1 is false for t=1 (B_{4,3}(3)=6), so the paper's headline (4,3^t) family is unsupported, but the other new families deserve a careful referee. read the letter →

arxiv 2507.01529 v4 pith:PNWMFH3J submitted 2025-07-02 math.NT

N. K. Meher , Sourav Bhowmick This is my paper
classification math.NT MSC 05A1711P8311F11
keywords biregularoverpartitionscongruenceseta-quotientsHeckeeigenformsdissectionformulasmodularformspartitionarithmeticprogressions
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

The paper studies $\bar{B}_{\ell_1,\ell_2}(n)$, the number of overpartitions of $n$ with no part divisible by $\ell_1$ or $\ell_2$ for coprime $\ell_1,\ell_2>1$. It aims to prove that these counts vanish along infinitely many arithmetic progressions: modulo 8 for pairs $(2,9)$ and $(5,2^t)$, modulo 4 for $(5,2)$, $(5,4)$, and $(3,2^t)$, and modulo 3 for $(2,9)$ and $(8,3)$, with the principal advertised family being $\bar{B}_{4,3^t}(3n)\equiv 0 \pmod{8}$ for every $t\geq 1$. The proofs isolate residue-class generating functions, identify the resulting eta-quotients as Hecke eigenforms, and transfer coefficient vanishing under primes in suitable residue classes back to the original counts. The proof of the $(4,3^t)$ family rests on a reduction that is false at $t=1$, since the same generating function then begins $1+6q+16q^2+\cdots$ rather than $1 \pmod{8}$.

What carries the argument

The engine is the eta-quotient form of the generating function, combined with dissection formulas that isolate the coefficients in a fixed residue class. After extraction the paper obtains identities such as $\sum_{n\geq 0}\bar{B}_{2,9}(6n+1)q^n \equiv 2\eta^4(6z) \pmod{8}$ and $\sum_{n\geq 0}\bar{B}_{5,2^t}(4n+1)q^n \equiv 2\eta(4z)\eta(20z) \pmod{8}$. The products $\eta^4(6z)$ and $\eta(4z)\eta(20z)$ are cusp forms that the paper identifies as Hecke eigenforms, meaning eigenvalues of the Hecke operators; applying $T_p$ at primes $p$ with vanishing eigenvalue forces certain coefficients $a(pn)+\cdots$ to vanish, which translates into zero congruence classes for the partition function. A second mechanism, a classical multiplicative identity for eta-products due to Newman [19], iterates coefficient relations at primes $p\equiv 1 \pmod{6}$ or $p\equiv 1 \pmod{4}$ to generate additional infinite families.

What would settle it

Evaluate the $t=1$ case of the claimed $(4,3^t)$ family: expanding the generating function for $(\ell_1,\ell_2)=(4,3)$ gives $\sum_{n\geq 0}\bar{B}_{4,3}(3n)q^n = 1+6q+16q^2+\cdots$, so $\bar{B}_{4,3}(3)=6$ and the asserted congruence $\bar{B}_{4,3}(3n)\equiv 0 \pmod{8}$ fails at $n=1$. This coefficient calculation settles the $t=1$ case directly.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is a collection of infinite congruence families for $\bar{B}_{\ell_1,\ell_2}(n)$. For primes $p_i\not\equiv 1 \pmod{6}$ it claims congruences of the form $\bar{B}_{2,9}(6p_1^2\cdots p_{k+1}^2 n + (6j+p_{k+1})p_1^2\cdots p_k p_{k+1})\equiv 0 \pmod{8}$ with an analogous family modulo 3, and multiplicative recurrence formulas for the same sequence. For $p_i\not\equiv 1 \pmod{4}$ it claims the same progression shape for $\bar{B}_{5,2^t}(4p_1^2\cdots p_{k+1}^2 n + (4j+p_{k+1})p_1^2\cdots p_k p_{k+1})\equiv 0 \pmod{8}$ for $t\geq 3$, with the modulo-4 analogue for $(5,2)$ and $(5,4)$. It also claims $\bar{B}_{8,3}(36n+33)\equiv 0 \pmod{3}$, $\bar{B}_{3,2^t}(16n+6)\equiv \bar{B}_{3,2^t}(16n+10)\equiv \bar{B}_{3,2^t}(16n+14)\equiv 0 \pmod{8}$, and, for $(4,3^t)$, $\bar{B}_{4,3^t}(3n)\equiv 0 \pmod{8}$ for all $n\geq 1$ together with several modulo-4 families.

Load-bearing premise

The proof of the main $(4,3^t)$ family assumes that the reduced generating function $\sum_{n\geq 0}\bar{B}_{4,3^t}(3n)q^n$ is congruent to $1$ modulo $8$ for every $t\geq 1$; for $t=1$ the same function is $1+6q+16q^2+\cdots$, so that assumption fails already in the first case.

Editorial extensions

If this is right

  • For every prime $p\equiv 5 \pmod{6}$, the claimed families give $\bar{B}_{2,9}(6p^{2k+2}n+6p^{2k+1}j+p^{2k+2})\equiv 0 \pmod{8}$ and the analogous modulo-3 family with $18$ in place of $6$, whenever $p\nmid j$.
  • For every prime $p\equiv 3 \pmod{4}$, the claimed families give $\bar{B}_{5,2^t}(4p^{2k+2}n+4p^{2k+1}j+p^{2k+2})\equiv 0 \pmod{8}$ for every $t\geq 3$, and the same progression shape modulo 4 for $(5,2)$ and $(5,4)$.
  • The paper claims $\bar{B}_{8,3}(36n+33)\equiv 0 \pmod{3}$ for all $n$, and $\bar{B}_{3,2^t}(16n+r)\equiv 0 \pmod{8}$ for $r=6,10,14$ and every $t\geq 1$.
  • The paper claims $\bar{B}_{4,3^t}(3n)\equiv 0 \pmod{8}$ for all $n\geq 1$ and all $t\geq 1$, together with $\bar{B}_{4,3^t}(3n+2)\equiv \bar{B}_{4,3^t}(6n+4)\equiv \bar{B}_{4,3^t}(12n+7)\equiv 0 \pmod{4}$.
  • If the claimed families hold, then each listed pair has positive arithmetic density of zero residues modulo the relevant power of 2 or modulo 3, a stronger property than isolated congruences.

Reading between the lines

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

  • Because the $t=1$ counterexample breaks only the reduction in equation (8.10), the natural repair is to restrict the $(4,3^t)$ family to $t\geq 2$; the modulo-4 portions of Theorem 8.1 derive from a different generating function and may survive unchanged.
  • The same template of reducing a residue class to an eta-product, identifying it as a Hecke eigenform, and iterating the eigenvalue recursion should produce analogous families for other coprime pairs $(\ell_1,\ell_2)$ whenever the reduced product is a low-weight newform with only finitely many bad primes.
  • The multiplicative recurrences stated for $(2,9)$ and $(5,2^t)$ give a cheap numerical test of any corrected $(4,3^t)$ conjecture: compute $\bar{B}_{4,3^t}(3p)$ for small primes $p$ and compare with the recurrence predicted by the would-be eigenform.
  • The paper's methods could be pushed further to seek congruences for the missing residue classes in the $(8,3)$ case, where only a single progression modulo 3 is established, rather than an infinite family.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper studies the arithmetic of (ℓ1,ℓ2)-biregular overpartitions, i.e. overpartitions with no part divisible by ℓ1 or ℓ2, denoted B̄_{ℓ1,ℓ2}(n). Using generating function manipulations, dissection formulas, Hecke eigenforms, and Newman's identity, the author claims infinitely many congruence families modulo 3 and powers of 2 for the pairs (2,9), (5,2), (5,4), (8,3), and general families (5,2^t) for t≥3, (3,2^t) for t≥1, and (4,3^t) for t≥1. The main advertised new results are Theorem 8.1, asserting congruences for B̄_{4,3^t}(n) modulo 8 and 4 for all t≥1, and Theorem 9.1, asserting congruences for B̄_{3,2^t}(n) modulo 8 for all t≥1.

Significance. If the claimed infinite families were correct, they would constitute a meaningful extension of recent results on biregular overpartitions. The paper is self-contained in its use of standard tools: no free parameters are fitted, and the proofs are derivations from cited external results such as Martin's classification of multiplicative eta-quotients, Newman's identity, and classical dissection formulas. However, the significance of the current version is severely undermined because two of the headline theorems are false as stated, and the abstract advertises precisely those families.

major comments (2)
  1. [Section 8, Eq. (8.10)] The congruence asserted in (8.10), namely ∑_{n≥0} B̄_{4,3^t}(3n)q^n ≡ 1 (mod 8) for all t≥1, is false. For t=1, the right-hand side of (8.7) expands as 1+6q+O(q^2), so B̄_{4,3}(3)=6, which is not congruent to 0 mod 8. Hence (8.2) fails for n=1, and Theorem 8.1 is false as stated. The proof's appeal to Lemma 2.8 does not justify (8.10), because the eta-quotient in (8.7) is not of the form appearing in (2.12); the reduction to 1 mod 8 is a non-sequitur. Since the abstract explicitly promises the (4,3^t) family for all t≥1, this is a load-bearing error in the paper's central claim.
  2. [Section 9, Theorem 9.1] Theorem 9.1 is false for t=1. Direct enumeration of overpartitions of 6 with no part divisible by 2 or 3 gives B̄_{3,2}(6)=6: the partitions 5+1 and 1+1+1+1+1+1 contribute 4 and 2 overpartitions respectively. Thus the assertion (9.2) with n=0, namely B̄_{3,2}(6)≡0 (mod 8), fails. Moreover, the derivation leading to (9.9) involves f_{2^{t-2}}, which is undefined for t=1, so the proof does not cover the stated range t≥1. The advertised (3,2^t) family for all t≥1 is therefore unsupported and false as stated.
minor comments (5)
  1. [Section 3, Proposition 1] The opening sentence says 'By setting (ℓ1, ℓ2) = (4,3) in (1.1)', but the generating function that follows is for B̄_{2,9}(n); the pair should be (2,9).
  2. [Section 4, Eq. (4.11)] Equation (4.11) states the congruence for B̄_{5,2}(4n+1), but the theorem concerns B̄_{5,2^t}(4n+1); this appears to be a typographical error.
  3. [Section 6, Proposition 4] The proof of Proposition 4 is labeled 'Proof of Proposition 5.3', which is an incorrect cross-reference.
  4. [Section 2, Lemma 2.8] The statement of Lemma 2.8 is typeset ambiguously; the intended congruence f_{p^{k-1}m}^p ≡ f_{p^k m} (mod p^k) should be written explicitly to avoid confusion.
  5. [Throughout] There are numerous typographical and grammatical issues (e.g., 'partions', 'notaion', 'expnasion', 'Devolepments') that should be corrected in any revision.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the congruences are derived from external q-series dissections and modular-form results, not from self-referential inputs or fitted predictions.

full rationale

The paper's derivation chain reduces the target generating functions through standard dissection formulas (Lemmas 2.5–2.10), the eta-quotient integrality and cusp-order criteria (Theorems 2.3–2.4), Martin's classification of multiplicative eta-quotients [17] (used to identify eta-quotients such as η^4(6z) and η(4z)η(20z) as Hecke eigenforms), and Newman's identity [19]. None of these inputs is defined in terms of the target coefficients B_{ℓ1,ℓ2}(n), and no free parameter is fitted to a subset of those coefficients and then reported as a prediction. The only author-overlapping reference, [28], is listed but not used in any proof. The eigenform and Newman results are external mathematical theorems, not self-citations that carry the argument on their own authority. The q-series manipulations are ordinary extractions of powers of q and applications of congruences for eta-quotients, so the resulting congruences are not assumed by construction. The apparent failure of (8.10) for t = 1, with B_{4,3}(3) = 6, is a correctness defect in an intermediate reduction, not a circular step: it does not define the target congruence in terms of itself, fit a parameter to data, or import a uniqueness theorem from the authors' own prior work. Therefore the paper exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard modular-form machinery and cited eigenform and Newman results. No free parameters are fitted. The failure of the t=1 reduction is not circularity but a computational error in applying Lemma 2.12.

assumptions (6)
  • domain assumption The eta-quotient \eta(6z)^4 is a Hecke eigenform of weight 2 and level 36.
    Cited to Martin [17]; used in Theorems 3.1 and 3.5 to obtain Hecke eigenvalue relations for the coefficients a(n).
  • domain assumption The eta-quotient \eta(4z)\eta(20z) is a Hecke eigenform of weight 1 and level 80.
    Cited to Martin [17]; used in Theorems 4.1, 5.1, and 6.1.
  • domain assumption Newman's identity gives multiplicative recurrences for coefficients of f_1 f_3 and f_1 f_5.
    Cited to Newman [19]; used in Theorems 3.9 and 4.5.
  • standard math The dissection formulas in Lemmas 2.5 through 2.10 are valid.
    These are standard q-series identities from Berndt, Hirschhorn, Xia and Yao, and others, used throughout the proofs.
  • standard math Lemma 2.12: for primes p and integers k,m\ge 1, f_{p^{k-1}m}^p \equiv f_{p^k m} (mod p^k).
    Cited to [26]; used to reduce eta-quotients modulo powers of 2 and 3.
  • standard math Theorems 2.3 and 2.4 on eta-quotients becoming modular forms on \Gamma_0(N).
    Standard modular forms background from Ono's book, used to prove membership in S_k(\Gamma_0(N),\chi).

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Cite this review

Pith. "Pith review of New Congruences on Biregular Overpartitions." pith.science (2026). https://pith.science/paper/PNWMFH3J

@misc{pith2026250701529,
  author       = {Pith},
  title        = {Pith review of: New Congruences on Biregular Overpartitions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PNWMFH3J}},
  note         = {Machine review of arXiv:2507.01529}
}
abstract

Recently, Nadji, Ahmia and Ram\'{i}rez \cite{Nadji2025} investigated the arithmetic properties of ${\bar B}_{\ell_1,\ell_2}(n)$, the number of overpartitions where no part is divisible by $\ell_1$ or $\ell_2$ with $\gcd(\ell_1,\ell_2)$$=1$ and $\ell_1$, $\ell_2>1$. Specifically, they established congruences modulo $3$ and powers of $2$ for the pairs $(\ell_1, \ell_2)$ $\in$ $\{(4,3),(4,9),(8,3),(8,9)\}$, using the concept of generating functions, dissection formulas and Smoot's implementation of Radu's Ramanujan-Kolberg algorithm. Further, Alanazi, Munagi and Saikia \cite{Alanazi2024} established some congruences for the pairs $(\ell_1,\ell_2)$ $\in$ $\{(2,3),(4,3),(2,5),(3,5),(4,9),(8,27),(16,81)\}$ using the theory of modular forms and Radu's algorithm. Recently, Paudel, Sellers and Wang \cite{Paudel2025} extended several of their results and established infinitely many families of new congruences. In this paper, we find infinitely many families of congruences modulo $3$ and powers of $2$ for the pairs $(\ell_1,\ell_2)$ $\in$ $ \{(5,2^t), (4,3^t)\}$ $\forall t\geq1$ with $t\in\mathbb{N}$ and for $(3,2^t)$ $\forall t\geq2 $ with $t\in\mathbb{N}$, using the theory of Hecke eigenforms, an identity due to Newman \cite{Newman1959}, the concept of dissection formulas.

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Forward citations

Cited by 1 Pith paper

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

  1. On Some New Congruences For Biregular Overpartitions

    math.NT 2025-07 conditional novelty 5.0 of 10

    B_{2^alpha,3^beta}(n), the number of overpartitions whose parts avoid multiples of 2^alpha and 3^beta, satisfies new congruence families modulo 4, 8, 6, and 12.

Reference graph

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