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Some congruences for $(s,t)$-regular bipartitions modulo $t$

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Pith's one-line read The paper proves four infinite families of congruences modulo 5, 11, and 17 for the numbers of $(s,t)$-regular bipartitions, including scaled recurrences and vanishing along arithmetic progressions.

desk verdict A narrow but solid congruence paper; the unshown Section 4 block checks out, so it deserves a serious referee. read the letter →

arxiv 1908.06642 v2 pith:NSG2FBU5 submitted 2019-08-19 math.NT

T. Kathiravan , K. Srilakshmi This is my paper · ORCID
classification math.NT MSC 11P8305A17
keywords CongruenceRegularbipartitionPartitionfunctionGeneratingq-seriesidentityDissectionModulo11Infinitefamily
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 establishes four new infinite families of congruences for $(s,t)$-regular bipartitions, the number $B_{s,t}(n)$ of bipartitions of $n$ whose first part is $s$-regular and second part is $t$-regular. Theorem 1.1 gives congruences modulo $5$ for $B_{2,15}$, including the exact scaling law $B_{2,15}(3^{2m+1}n+(7\cdot 3^{2m+1}-5)/8)\equiv 2^m B_{2,15}(3n+2)\pmod 5$. Theorem 1.2 gives a scaling law modulo $11$ for $B_{7,11}$ and shows that $B_{7,11}$ vanishes modulo $11$ along three families of arithmetic progressions. Theorem 1.3 shows that $B_{27,11}$ vanishes modulo $11$ for all $m\ge 4$, and Theorem 1.4 shows that $B_{243,17}$ vanishes modulo $17$ in two residue classes modulo $81$. These results extend a line of congruence results for regular bipartitions and give infinitely many explicit values of $n$ for which each function is divisible by the relevant modulus.

What carries the argument

The argument runs on the generating function $\sum_{n\ge0}B_{s,t}(n)q^n=f_sf_t/f_1^2$, with $f_k=\prod_{m\ge1}(1-q^{mk})$. Since $f_t\equiv f_1^t\pmod t$ by the binomial theorem, the problem reduces to extracting coefficients from products of $f_1$'s and a single $f_t$. The dissections use classical identities: Berndt's identity (2.2), the Hirschhorn--Sellers identity (2.3), and Hirschhorn's cube and reciprocal-cube identities (2.4)--(2.6), together with Lemmas 2.4--2.6 for $p_5(7n+3)$, $p_7(7n)$, and $p_9(7n+4)$ built from $\theta$-function relations from Ramanujan's lost notebook. Extracting terms in a fixed residue class modulo $3$ or $7$, then replacing $q^3$ or $q^7$ by $q$, produces the recurrences that iterate to Theorems 1.1--1.4.

What would settle it

Evaluate the generating function $\sum B_{7,11}(n)q^n=f_7f_{11}/f_1^2$ to the first few hundred terms and check the residue of $B_{7,11}((2\cdot7^{12}-2)/3)$ against $3\pmod{11}$, and check $B_{7,11}(7^{11}+(2\cdot7^{11}-2)/3)\equiv0\pmod{11}$; a single mismatch would disprove Theorem 1.2.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the generating function $\sum_{n\ge0}B_{s,t}(n)q^n=f_sf_t/f_1^2$ can be dissected, after reduction modulo $t$ via $f_t\equiv f_1^t\pmod t$, to yield exact recurrences between values of $B_{s,t}$ at arithmetic progressions whose step multiplies by $3$ or $7$ at each iteration. For $(2,15)$ and $(7,11)$ the recurrences are nonzero scaling laws: $B_{2,15}(3^{2m+1}n+(7\cdot3^{2m+1}-5)/8)\equiv2^mB_{2,15}(3n+2)\pmod5$ and $B_{7,11}(7^{12m}n+(2\cdot7^{12m}-2)/3)\equiv3^mB_{7,11}(n)\pmod{11}$. For $(7,11)$ with $k=1,5,6$, for $(27,11)$ with $m\ge4$, and for $(243,17)$ in the residue classes $23$ and $77$ modulo $81$, the same method yields outright vanishing modulo the modulus. All four families are asserted for all $n\ge0$ and all permitted $m$, so the paper claims infinitely many congruence identities, not merely finitely many checked cases.

Load-bearing premise

The proofs of Theorem 1.2 depend on a block of ten dissection formulas in equation (4.7) that the paper asserts with the phrase 'Similarly, we find' rather than proving line by line; a single wrong coefficient there would change the residues used to obtain the congruences.

Editorial extensions

If this is right

  • For $(2,15)$-regular bipartitions, the paper proves that $B_{2,15}$ vanishes modulo $5$ on the progressions $3^{2m+2}n+(23\cdot3^{2m+1}-5)/8$ and $3^{2(m+1)+1}n+(13\cdot3^{2(m+1)}-5)/8$ for every $m\ge0$.
  • For $(7,11)$-regular bipartitions, $B_{7,11}(7^{12m}n+(2\cdot7^{12m}-2)/3)\equiv3^mB_{7,11}(n)\pmod{11}$ for all $m\ge0$, so the same residue recurs with a $3^m$ multiplier.
  • For $(7,11)$-regular bipartitions, $B_{7,11}(7^{12m+11}(7n+k)+(2\cdot7^{12m+11}-2)/3)\equiv0\pmod{11}$ for $k=1,5,6$ and all $m,n\ge0$.
  • For $(27,11)$-regular bipartitions, $B_{27,11}(3^m n+(5\cdot3^{m-1}-3)/2)\equiv0\pmod{11}$ for all $m\ge4$ and $n\ge0$, and for $(243,17)$-regular bipartitions, $B_{243,17}(81n+23)\equiv B_{243,17}(81n+77)\equiv0\pmod{17}$ for all $n\ge0$.

Reading between the lines

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

  • The same binomial reduction and dissection recipe should apply to other pairs $(s,t)$ with $t$ prime once $f_sf_t/f_1^2$ collapses to a manageable product modulo $t$; the four pairs here display the pattern for $t=5,11,17$.
  • The $(243,17)$ congruences are proved only at the base scale, so a natural open question is whether the same two residue classes recur at higher powers of $3$, as the other theorems' base congruences do.
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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

1 major / 3 minor

Summary. The paper studies B_{s,t}(n), the number of (s,t)-regular bipartitions of n, whose generating function is f_s f_t / f_1^2. Using modular q-series dissections and known identities from Ramanujan, Berndt, Hirschhorn, and Sellers, the authors prove four theorems: Theorem 1.1 gives a two-parameter family of congruences modulo 5 for B_{2,15}; Theorem 1.2 gives infinite families modulo 11 for B_{7,11}, including a strong congruence with a power of the modulus in the argument; Theorem 1.3 gives a family modulo 11 for B_{27,11} for all m ≥ 4; and Theorem 1.4 gives two residue classes modulo 17 for B_{243,17}. The proofs proceed by extracting coefficients from generating functions, applying dissections, and iterating the resulting recurrences.

Significance. The paper provides new infinite families of congruences for four families of regular bipartition functions, extending a recent line of results by Lin, Dou, Xia and Yao, Adiga and Ranganatha, and Kathiravan. The methods are standard but the computations are explicit, checkable, and free of fitted parameters. The final congruences are concrete and falsifiable. The main families in Theorems 1.1 and 1.2 are particularly clean, and the paper would be a useful addition to the literature once the omitted derivation in Section 4 is supplied.

major comments (1)
  1. [Section 4, between (4.6) and (4.7)] The ten iterative dissections displayed after the phrase 'Similarly, we find' are stated without any derivation. This block is essential: the last line of (4.7) is substituted into (2.9) to obtain (4.8) and hence (4.9)-(4.10), so Theorem 1.2 rests on the correctness of these ten lines. I have checked the block independently: writing R_k = A f_7 f_1^9 + B q f_7^5 f_1^5 + C q^2 f_7^9 f_1, the operation of extracting the residue class 4 modulo 7 gives (A',B',C') ≡ (9A+10B+10C, 9A+5B, 8A) modulo 11, and starting from (9,9,8) this reproduces exactly the coefficient triples (9,5,6), (4,7,6), (1,5,10), (5,1,8), (3,6,7), (3,2,2), (1,4,2), (3,7,8), (1,7,2), (0,0,8) in (4.7). The mathematics is therefore correct, but as written the proof is not self-contained in a load-bearing place. The authors should include the recurrence, or at least one full representative iteration with the remaining lines listed as the result of repeating it.
minor comments (3)
  1. [Section 6, after (6.5)] The final sentence 'This completes the proof of Theorem 1.4 follow from (6.5)' omits the key observation that every term on the right-hand side of (6.5) is a power series in q^3 multiplied by q, so the coefficients of q^{3n} and q^{3n+2} vanish. This extraction should be stated explicitly.
  2. [Section 5, proof of Theorem 1.3] The passage from (5.7) and (5.8) to the full infinite family (1.12) is not shown. A one-line induction, with base case m=4 given by (5.7) and the induction step following from (5.8), should be included.
  3. [Throughout] There are numerous typographical and grammatical errors, such as 'Ramanujan [18] give', 'th e identities', 'mo dulo', 'regu lar', and 'co efficient'. A careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: proofs are self-contained and rely on external identities; target congruences are never assumed, and self-citations are contextual only.

full rationale

The paper's central claims are congruences for B_{s,t}(n). Every proof starts from the standard generating function (1.6), reduces it modulo t via the binomial fact f_p ≡ f_1^p (mod p), and then uses coefficient extraction together with published external identities: Berndt (2.2), Hirschhorn and Sellers (2.3), Hirschhorn (2.4)-(2.6), and Berndt-Yee-Yi (2.12)-(2.15). The in-paper lemmas (2.4) and (2.6) are derived by substituting these external identities, not by assuming any B_{s,t} congruence. Theorem 1.2's delicate block marked 'Similarly, we find' between (4.6) and (4.7) is an omitted computation, not a circular one: its final line is used only after coefficient extraction, and the printed coefficients are independently checkable from the recurrence on the (A,B,C) triple. The only self-citations, [14] and [15], appear in the introduction as literature context and no theorem from either is load-bearing in the proofs of Theorems 1.1-1.4. No fitted parameters, no renamed empirical pattern, and no definition that presupposes the target result occur. The derivation is therefore self-contained relative to the external identities it invokes.

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

The proof leans entirely on cited q-series identities and standard generating-function manipulations. No free parameters, fitted values, or new entities are introduced. The central congruences are derived, not assumed.

assumptions (6)
  • standard math Binomial congruence f_p is congruent to f_1^p modulo p for prime p (equation (2.1)).
    Used throughout to replace f_t by f_1^? modulo t and to reduce coefficients modulo primes. Standard consequence of the binomial theorem.
  • domain assumption Berndt's identity f_2^2/f_1 = f_6 f_9^2/(f_3 f_18) + q f_18^2/f_9 (Lemma 2.1, cited to Berndt).
    Used in Section 3 for the (2,15) dissection. The result is taken from published literature, not proved here.
  • domain assumption Hirschhorn-Sellers identity f_2/f_1^2 = f_6^4 f_9^6/(f_3^8 f_18^3) + 2q f_6^3 f_9^3/f_3^7 + 4q^2 f_6^2 f_18^3/f_3^6 (Lemma 2.2).
    Used in Section 3 for the (2,15) generating function. Taken from published literature.
  • domain assumption Hirschhorn's identities (2.4)-(2.6) involving f_1^3, a(q), and 1/f_1^3 (Lemma 2.3).
    Used in Sections 5 and 6 for (27,11) and (243,17). a(q) is the two-variable theta sum; these identities are cited to Hirschhorn.
  • domain assumption Berndt's modular equation f_1 = f_49 (B(q^7)/C(q^7) - A(q^7)/B(q^7) q - q^2 + C(q^7)/A(q^7) q^5), with A, B, C defined (equation (2.9)).
    Used in Lemmas 2.4 and 2.6 and in Section 4 to extract residues modulo 7. Cited to Berndt's Notebooks, Part III, Entry 17(v).
  • domain assumption Berndt-Yee-Yi identities (2.12)-(2.15) relating A, B, C and f_1, f_7.
    Used in the proofs of Lemmas 2.4 and 2.6, and indirectly in Section 4. Cited to Berndt-Yee-Yi (2003).

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Pith. "Pith review of Some congruences for $(s,t)$-regular bipartitions modulo $t$." pith.science (2026). https://pith.science/paper/NSG2FBU5

@misc{pith2026190806642,
  author       = {Pith},
  title        = {Pith review of: Some congruences for $(s,t)$-regular bipartitions modulo $t$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NSG2FBU5}},
  note         = {Machine review of arXiv:1908.06642}
}
abstract

In this work, we study the function $B_{s,t}(n)$, which counts the number of $(s,t)$-regular bipartitions of $n$. Recently, many authors proved infinite families of congruences modulo $11$ for $B_{3,11}(n)$, modulo $3$ for $B_{3,s}(n)$ and modulo $5$ for $B_{5,s}(n)$. Very recently, Kathiravan proved several infinite families of congruences modulo $11$, $13$ and $17$ for $B_{5,11}(n)$, $B_{5,13}(n)$ and $B_{81,17}(n)$. In this paper, we will prove infinite families of congruences modulo $5$ for $B_{2,15}(n)$, modulo $11$ for $B_{7,11}(n)$, modulo $11$ for $B_{27,11}(n)$ and modulo $17$ for $B_{243,17}(n)$.

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Reference graph

Works this paper leans on

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