Some New Congruences for $l$-Regular Partitions Modulo $l$

K. Srilakshmi; S. Abinash; T. Kathiravan

arxiv: 1907.08848 · v1 · pith:FKRJBZLGnew · submitted 2019-07-20 · 🧮 math.NT

Some New Congruences for l-Regular Partitions Modulo l

S. Abinash , T. Kathiravan , K. Srilakshmi This is my paper

Pith reviewed 2026-05-24 18:41 UTC · model grok-4.3

classification 🧮 math.NT

keywords l-regular partitionspartition congruencesRamanujan theta functionsgenerating functionsmodulo primesarithmetic progressions

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The pith

Ramanujan's theta function identities yield new infinite families of congruences for the count of l-regular partitions modulo l when l is 13, 17 or 23.

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

The paper aims to establish new congruences satisfied by b_l(n), the number of l-regular partitions of n (partitions with no part divisible by l). It does this for the primes l=13,17,23 by rewriting the generating function in terms of Ramanujan's theta functions and transferring known coefficient congruences. A sympathetic reader would care because these results enlarge the set of arithmetic progressions in which the partition counts are forced to vanish modulo l. The work focuses on producing explicit infinite families rather than isolated cases.

Core claim

Using Ramanujan's theta function identities, the generating function for b_l(n) is rewritten so that its coefficients satisfy new congruences modulo l for each of l=13,17,23; this produces infinite families of arithmetic progressions in which b_l(n) is divisible by l.

What carries the argument

Ramanujan's theta function identities applied directly to the generating function of l-regular partitions, allowing coefficient congruences to transfer without correction terms.

If this is right

For each l in {13,17,23} there exist infinitely many residue classes where b_l(n) ≡ 0 mod l.
The families are obtained uniformly from the same theta-function manipulation for all three primes.
No level-lowering or extra modular-form arguments are required beyond the theta identities.

Where Pith is reading between the lines

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

The same rewriting technique could be tested on other small primes once suitable theta identities are identified.
The congruences may be used to study the distribution or asymptotic density of l-regular partitions inside arithmetic progressions.
Similar direct inheritance might hold for congruences modulo higher powers of l or for related partition statistics.

Load-bearing premise

The generating function for b_l(n) can be expressed so that it directly inherits the coefficient congruences already known for the theta functions.

What would settle it

An explicit integer n for which b_13(n) fails to satisfy one of the predicted congruences modulo 13.

read the original abstract

A partition of $n$ is $l$-regular if none of its parts is divisible by $l$. Let $b_l(n)$ denote the number of $l$-regular partitions of $n$. In this paper, using theta function identities due to Ramanujan, we establish some new infinite families of congruences for $b_l(n)$ modulo $l$, where $l=13,17,23$.

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

Paper adds three new families of congruences for b_l(n) mod l at l=13,17,23 via Ramanujan theta identities, but the mod-l reduction step on the generating function ratio needs explicit verification.

read the letter

The colleague should know that this paper states three new infinite families of congruences for the count of l-regular partitions modulo l, specifically when l equals 13, 17, or 23. That is the concrete output. They obtain the families by rewriting the generating function for b_l(n) with Ramanujan's theta identities and reading off the coefficient congruences that follow from those identities. The families are new for these l values and no earlier reference is cited that already contains them. The method itself is the standard one already used for other regular partition congruences, so the advance is in the specific cases rather than in technique. The paper does the basic job of naming the identities and writing down the resulting statements clearly. The soft spot is the one flagged in the stress-test note. The generating function equals the ordinary partition generating function divided by its version evaluated at q^l. When coefficients are taken modulo l, the denominator contributes only on multiples of l, and cross terms can appear. The abstract gives no sign that the authors performed an explicit cancellation or applied a level-lowering argument to guarantee the claimed congruences survive unchanged. If the full text also skips that check, the derivation is incomplete on that point. This work is aimed at the small group of researchers who follow Ramanujan-style congruences for regular partitions and their close relatives. A reader already comfortable with the theta-function toolkit will see the extension at once and can verify the arithmetic by hand. It is worth sending to peer review. The claims are narrow and checkable, so referees can settle whether the reduction step holds.

Referee Report

1 major / 0 minor

Summary. The paper claims to derive new infinite families of congruences satisfied by b_l(n), the number of l-regular partitions of n, modulo l for the primes l=13,17,23. The derivations rely on expressing the generating function for b_l(n) in terms of Ramanujan's theta functions and transferring known coefficient congruences from those identities.

Significance. If correct, the results enlarge the collection of explicit arithmetic progressions along which b_l(n) vanishes modulo l. Such families are of interest in the arithmetic theory of partitions, and the approach via classical theta identities is a standard and potentially efficient route when the generating-function manipulations are fully justified.

major comments (1)

[derivation of the main congruences (likely §3 or §4)] The central step rewrites the generating function ∏_{k≢0 mod l} (1-q^k)^{-1} = P(q)/P(q^l) and invokes Ramanujan theta identities to extract coefficient congruences mod l. The manuscript must supply the explicit reduction argument showing that the factor 1/P(q^l) does not introduce uncancelled cross terms that would obstruct the claimed families; without this, the inheritance of the theta-function congruences is not automatic.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for identifying a point where the justification of the main congruences can be strengthened. We address the single major comment below and will incorporate the requested clarification.

read point-by-point responses

Referee: The central step rewrites the generating function ∏_{k≢0 mod l} (1-q^k)^{-1} = P(q)/P(q^l) and invokes Ramanujan theta identities to extract coefficient congruences mod l. The manuscript must supply the explicit reduction argument showing that the factor 1/P(q^l) does not introduce uncancelled cross terms that would obstruct the claimed families; without this, the inheritance of the theta-function congruences is not automatic.

Authors: We agree that an explicit reduction argument is needed to justify why the factor 1/P(q^l) does not obstruct the claimed congruences. In the revised version we will add a short lemma (placed immediately after the generating-function identity) that expands 1/P(q^l) as a power series in q^l and shows, modulo l, that its coefficients multiply the theta-function series only in ways that preserve the arithmetic progressions already known to be congruent to zero. The argument uses the fact that all exponents in the denominator are multiples of l together with the support of the theta identities, so no uncancelled cross terms arise in the relevant residue classes. This makes the transfer of congruences fully rigorous. revision: yes

Circularity Check

0 steps flagged

No circularity; relies on independent external identities

full rationale

The paper's derivation invokes Ramanujan theta function identities (external, predating the work) to transfer coefficient congruences to the generating function for b_l(n). No self-definitional reductions, fitted inputs renamed as predictions, self-citation load-bearing steps, or ansatz smuggling appear in the provided abstract or description. The approach is self-contained against the cited external benchmarks, with no reduction of the claimed families to the paper's own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on Ramanujan's theta-function identities as background results; no free parameters, invented entities, or ad-hoc axioms are visible from the abstract.

axioms (1)

standard math Ramanujan's theta-function identities hold and their coefficient congruences transfer directly to the generating function of b_l(n).
Invoked in the abstract as the source of the new congruences.

pith-pipeline@v0.9.0 · 5598 in / 1228 out tokens · 17461 ms · 2026-05-24T18:41:40.766968+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

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Ahlgren and J

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Carlson and J

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Cui and N

S.-P. Cui and N. S. S. Gu, Arithmetic properties of ℓ-regular partitions, Adv. in Appl. Math. 51(4) (2013), 507–523

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Cui and N

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D. Tang, Congruences modulo powers of 5 for k-colored partitions, J. Number Theory 187 (2018), 198–214

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L. W ang, Congruences for 5-regular partitions modulo p owers of 5, Ramanujan J. 44(2) (2017), 343–358

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W ang, Arithmetic properties of 7-regular partition s, Ramanujan J

L. W ang, Arithmetic properties of 7-regular partition s, Ramanujan J. 47(1) (2018), 99–115

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W ang, Congruences modulo powers of 11 for some partit ion functions, Proc

L. W ang, Congruences modulo powers of 11 for some partit ion functions, Proc. Amer. Math. Soc. 146(4) (2018), 1515–1528

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G. N. W atson, Ramanujans Vermutung ¨ uber Zerf¨ allungs zahlen, J. Reine Angew. Math. 179 (1938), 97–128

work page 1938
[25]

J. J. W ebb, Arithmetic of the 13-regular partition func tion modulo 3, Ramanujan J. 25(1) (2011), 49–56

work page 2011
[26]

E. X. W. Xia, New inﬁnite families of congruences modulo 8 for partitions with even parts distinct, Electron. J. Combin. 21(4) (2014), Paper 4.8, 10

work page 2014
[27]

E. X. W. Xia, Congruences for some l-regular partitions modulo l, J. Number Theory 152 (2015), 105–117

work page 2015
[28]

E. X. W. Xia and O. X. M. Yao, Parity results for 9-regular partitions, Ramanujan J. 34(1) (2014), 109–117

work page 2014
[29]

E. X. W. Xia and O. X. M. Yao, A proof of Keith’s conjecture for 9-regular partitions modulo 3, Int. J. Number Theory 10(3) (2014), 669–674

work page 2014
[30]

O. X. M. Yao, New congruences modulo powers of 2 and 3 for 9 -regular partitions, J. Number Theory 142 (2014), 89–101

work page 2014
[31]

T. Y. Zhao, J. Jin, and O. X. M. Yao, Parity results for 11- , 13- and 17-regular partitions, Colloq. Math. 151(1) (2018), 97–109. Indian Institute of Science Education and Research Thiruva nanthapuram, Maruthamala P.O., Vithura, Thiruvananthapuram-695551, Kerala, India. E-mail address : sarmaabinash15@iisertvm.ac.in The Institute of Mathematical Sciences...

work page 2018

[1] [1]

Ahlgren and J

S. Ahlgren and J. Lovejoy, The arithmetic of partitions i nto distinct parts, Mathematika 48(1-2) (2001), 203–211

work page 2001

[2] [2]

G. E. Andrews, M. D. Hirschhorn, and J. A. Sellers, Arithm etic properties of partitions with even parts distinct, Ramanujan J. 23(1-3) (2010), 169–181

work page 2010

[3] [3]

B. C. Berndt, Ramanujan ’s Notebooks: Part III (Springer, New York, 1991)

work page 1991

[4] [4]

B. C. Berndt, Ramanujan ’s Notebooks: Part IV (Springer, New York, 1994)

work page 1994

[5] [5]

B. C. Berndt, A. J. Yee, and J. Yi, Theorems on partitions f rom a page in Ramanujan’s lost notebook, in Proceedings of the International Conference on Special Functions and th eir Applications, ed. R. Jagannathan, S. Kanemitsu, G. V. Berghe, and W. V. Assche, J. Comput. Appl. Math., Vol. 160( 1-2) (Elsevier, 2003), pp. 53–68

work page 2003

[6] [6]

Carlson and J

R. Carlson and J. J. W ebb, Inﬁnite families of inﬁnite fam ilies of congruences for k-regular partitions, Ramanujan J. 33(3) (2014), 14–23

work page 2014

[7] [7]

Cui and N

S.-P. Cui and N. S. S. Gu, Arithmetic properties of ℓ-regular partitions, Adv. in Appl. Math. 51(4) (2013), 507–523

work page 2013

[8] [8]

Cui and N

S.-P. Cui and N. S. S. Gu, Congruences for 9-regular parti tions modulo 3, Ramanujan J. 38(3) (2015), 503–512

work page 2015

[9] [9]

H. Dai, C. Liu, and H. Yan, On the distribution of odd value s of 2 a-regular partition functions, J. Number Theory 143 (2014), 14–23

work page 2014

[10] [10]

Dandurand and D

B. Dandurand and D. Penniston, l-divisibility of l-regular partition functions, Ramanujan J. 19(1) (2009), 63–70

work page 2009

[11] [11]

Furcy and D

D. Furcy and D. Penniston, Congruences for ℓ-regular partition functions modulo 3, Ramanujan J. 27(1) (2012), 101–108

work page 2012

[12] [12]

W. J. Keith, Congruences for 9-regular partitions modu lo 3, Ramanujan J. 35(1) (2014), 157–164

work page 2014

[13] [13]

B. L. S. Lin and A. Y. Z. W ang, Generalisation of Keith’s c onjecture on 9-regular partitions and 3-cores, Bull. Aust. Math. Soc. 90(2) (2014), 204–212

work page 2014

[14] [14]

Lovejoy and D

J. Lovejoy and D. Penniston, 3-regular partitions and a modular K3 surface, in q-Series with Applications to Com- binatorics, Number Theory, and Physics, ed. B. C. Berndt and K. Ono, Contemp. Math., Vol. 291 (Amer. Ma th. Soc., Providence, RI, 2001), pp. 177–182

work page 2001

[15] [15]

Penniston, The pa-regular partition function modulo pj, J

D. Penniston, The pa-regular partition function modulo pj, J. Number Theory 94(2) (2002), 320–325

work page 2002

[16] [16]

Penniston, Arithmetic of l-regular partition functions, Int

D. Penniston, Arithmetic of l-regular partition functions, Int. J. Number Theory 4(2) (2008), 295–302

work page 2008

[17] [17]

Ramanujan, in Collected Papers, ed

S. Ramanujan, in Collected Papers, ed. G. H. Hardy (New York, Chelsea, 1962)

work page 1962

[18] [18]

Tang, Congruences modulo powers of 5 for k-colored partitions, J

D. Tang, Congruences modulo powers of 5 for k-colored partitions, J. Number Theory 187 (2018), 198–214

work page 2018

[19] [19]

W ang, Congruences for 5-regular partitions modulo p owers of 5, Ramanujan J

L. W ang, Congruences for 5-regular partitions modulo p owers of 5, Ramanujan J. 44(2) (2017), 343–358

work page 2017

[20] [20]

W ang, Arithmetic properties of 7-regular partition s, Ramanujan J

L. W ang, Arithmetic properties of 7-regular partition s, Ramanujan J. 47(1) (2018), 99–115

work page 2018

[21] [21]

W ang, Congruences modulo powers of 11 for some partit ion functions, Proc

L. W ang, Congruences modulo powers of 11 for some partit ion functions, Proc. Amer. Math. Soc. 146(4) (2018), 1515–1528

work page 2018

[22] [22]

G. N. W atson, Theorems Stated by Ramanujan (IX) : Two Con tinued Fractions, J. London Math. Soc. 4(3) (1929), 231–237

work page 1929

[23] [23]

G. N. W atson, Theorems Stated by Ramanujan (VII): Theor ems on Continued Fractions, J. London Math. Soc. 4(1) (1929), 39–48

work page 1929

[24] [24]

G. N. W atson, Ramanujans Vermutung ¨ uber Zerf¨ allungs zahlen, J. Reine Angew. Math. 179 (1938), 97–128

work page 1938

[25] [25]

J. J. W ebb, Arithmetic of the 13-regular partition func tion modulo 3, Ramanujan J. 25(1) (2011), 49–56

work page 2011

[26] [26]

E. X. W. Xia, New inﬁnite families of congruences modulo 8 for partitions with even parts distinct, Electron. J. Combin. 21(4) (2014), Paper 4.8, 10

work page 2014

[27] [27]

E. X. W. Xia, Congruences for some l-regular partitions modulo l, J. Number Theory 152 (2015), 105–117

work page 2015

[28] [28]

E. X. W. Xia and O. X. M. Yao, Parity results for 9-regular partitions, Ramanujan J. 34(1) (2014), 109–117

work page 2014

[29] [29]

E. X. W. Xia and O. X. M. Yao, A proof of Keith’s conjecture for 9-regular partitions modulo 3, Int. J. Number Theory 10(3) (2014), 669–674

work page 2014

[30] [30]

O. X. M. Yao, New congruences modulo powers of 2 and 3 for 9 -regular partitions, J. Number Theory 142 (2014), 89–101

work page 2014

[31] [31]

T. Y. Zhao, J. Jin, and O. X. M. Yao, Parity results for 11- , 13- and 17-regular partitions, Colloq. Math. 151(1) (2018), 97–109. Indian Institute of Science Education and Research Thiruva nanthapuram, Maruthamala P.O., Vithura, Thiruvananthapuram-695551, Kerala, India. E-mail address : sarmaabinash15@iisertvm.ac.in The Institute of Mathematical Sciences...

work page 2018