Numérisation de Documents Anciens Mathématiques

Abraham, Uri

Pouzet, Maurice

The chain covering number of a poset with no infinite antichains

Bonin, Nathan

Ovsienko, Valentin

Comptes Rendus
Mathématique
Nathan Bonin and Valentin Ovsienko
A shadow Markov equation
Volume 361 (2023), p. 1483-1489
https://doi.org/10.5802/crmath.496
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1483-1489
https://doi.org/10.5802/crmath.496
Combinatorics / Combinatoire
A shadow Markov equation
Nathan Bonin a and Valentin Ovsienko ∗,a
a Laboratoire de Mathématiques de Reims, UMR9008 CNRS, Université de Reims Champagne-Ardenne,
U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse - BP 1039, 51687 Reims cedex 2, France
E-mails: nathan.bonin1@etudiant.univ-reims.fr, valentin.ovsienko@univ-reims.fr
Abstract. We introduce an analogue of the classical Markov equation that involves dual numbers a+αε with
ε2 = 0. This equation characterizes the “shadow Markov numbers” recently considered by one of us. We show
that this equation is characterized by invariance by cluster algebra mutations.
Funding. This project was partially supported by the ANR project ANR-19-CE40-0021.
Manuscript received 9 December 2022, revised 2 March 2023 and 8 March 2023, accepted 12 March 2023.
1. Introduction
The classical Markov (or Markoff) equation [6] is the Diophantine equation
a2 +b2 + c2 = 3abc.
A positive integer a is called a Markov number if it can be completed to a triplet of positive in-
tegers (a,b,c) satisfying the Markov equation. Markov numbers attracted much interest in many
branches of mathematics, from number theory to topology, combinatorics, and mathematical
physics.
In this note we will consider pairs of positive integers, (a,α) ∈ Z×Z, that will be written in a
form
A = a +αε, (1)
where ε is a formal parameter such that ε2 = 0. Formal sums (1) are called dual numbers. The
notion of dual number goes back to Clifford and is used in various situations in algebra, geometry,
and mathematical physics. A dual number (1) is called a positive integer if a,α ∈Z>0.
Our goal is to introduce and study a deformation of the Markov equation
A2 +B 2 +C 2 = (3−2ε) ABC . (2)
Solutions (A,B ,C ) to this equation are triples of dual numbers. We call a dual Markov number a
positive integer dual number A that can be completed to a triplet (A,B ,C ) of positive integer dual
numbers satisfying (2).
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1484 Nathan Bonin and Valentin Ovsienko
The following triplets of dual numbers
(1,1+ε,1+ε), (1+ε,1+ε,2+4ε), (1+ε,2+4ε,5+13ε)
are examples of triplets of dual Markov numbers.
We study several properties of the deformed Markov equation (2). The main property is that it
is preserved by cluster algebra mutations, similarly to the classical case [11]. We will show that it
is characterized by this mutation invariance.
The notion of “shadow” sequences of integers appeared in [8], and was tested on the sequence
of Markov numbers. Every Markov number, a, is accompanied by another positive integer, α,
called the shadow of a. For a definition, see [8] and Section 2. It is a priori not clear, and was
asked in [8], if the sequence of shadows of Markov numbers satisfy any equation. It turns out that
the classical Markov numbers organized together with their shadows in the form of dual numbers
are solutions to (2).
Theorem 1. Given a Markov number a and its shadow α, the corresponding dual number A =
a +αε is a dual Markov number.
We do not have a classification of dual Markov numbers. A classical theorem Markov implies
that every solution (A,B ,C ) to (2) can be obtained by a sequence of transformations called
mutations (see Section 2) from a triplet of the form
A′ = 1+α′ε, B ′ = 1+β′ε, C ′ = 1+γ′ε,
with some α′,β′,γ′ ∈ Z, but it is an open problem to determine the conditions that guarantee
positivity of α,β,γ.
Another open problem is to understand whether the equation (2) is connected to supergeom-
etry. We believe that (2) should be considered as a “superanalogue” of the classical Markov equa-
tion and is related to the supergroup OSp(1|2,Z); see [1]. However, we were unable to prove this
so far. One also notices a certain resemblance of (2) with the super-Markov equation of [5]; see
Section 3.2.
Although the proofs of the statements of this note are rather elementary, we hope that equa-
tion (2) can be useful for what is sometimes called in mathematical physics “super number the-
ory”; see [1,12]. The idea of shadow integer sequences recently initiated in [10] and [8] was further
developed in [1, 4, 14] and generated various unexpected connections.
2. Markov numbers and their shadows
In this section we briefly explain the connection of Markov numbers with Fomin–Zelevinsky
cluster algebras [2], that was proposed in [11]. We also recall the construction of the shadow
Markov sequence introduced in [8].
2.1. Markov numbers and quiver mutations
It is well-known and due to Markov [6], that every positive integer solution to the Markov equation
can be obtained from (1,1,1) by a sequence of transformations (a,b,c) 7→ (a′,b,c), where
a′ = b
2 + c2
a
, (3)
and permutations of a,b, and c. It is easy to check that (a′,b,c) remains a solution if (a,b,c) is a
solution.
Nathan Bonin and Valentin Ovsienko 1485
It was noticed in [11], that the maps (3) can be understood as instances of cluster algebra
mutations in the sense of Fomin–Zelevinsky cluster algebra [2]. The quiver associated with the
sequence of Markov numbers is as follows:
b

OO OO
a coooo
and (3) are precisely the exchange relations for the mutations associated to this quiver.
Integrality of a′ in (3) follows from the Laurent phenomemon [3], and can also be obtained
directly from an equivalent (and more frequently used) form of (3): a′ = 3bc −a.
2.2. The construction of [8]
The notion of shadow Markov sequence [8] is based on transformations (3). The shadow se-
quence is constructed recurrently without any equation on it.
Choose the initial triplet of dual numbers (A1,B1,C1) with
A1 = 1+α1ε, B1 = 1+β1ε, C1 = 1+γ1ε, (4)
where α1,β1,γ1 are arbitrary integers. Define an infinite sequence of transformations (A,B ,C ) 7→
(A′,B ,C ) by the same formula as (3):
A′ := B
2 +C 2
A
. (5)
More explicitly (5) reads
a′ = b
2 + c2
a
, α′ = 2bβ+2c γ−a
′α
a
(6)
and therefore determines both, the real and nilpotent parts of A′ = a′+α′ε. Furthermore, α′ will
remain integer, which follows from the version of Laurent phenomenon proved in [9].
2.3. The Markov tree and its shadow
Similarly to the classical Markov numbers, their shadows can be organized with the help of an
infinite binary tree. The version that we use is the following.
The binary tree is drawn in the plane cutting it into infinitely many regions, and every region
is labeled by a Markov number. Locally the picture is this
A B
•
C
and the transformations (5) correspond to the following branchings:
B
A • • A′
C
The most natural choice of the initial values (α1,β1,γ1) is
(α1,β1,γ1) = (0,1,1). (7)
1486 Nathan Bonin and Valentin Ovsienko
For details; see [8] and Section 4. The tree of Markov numbers with the initial triplet (7) is
presented below together with their shadow:
1 0 • 1 1
•
1 1 2 4
•
• 5 13 •
• 13 40 • • 29 117 •
• 34 • • 194 • • 433 • • 169 •
120 976 2592 921
89 1325 7561 2897 6466 37666 14701 985
354 7875 56287 20226 51320 352360 129640 6761
. . . . . . . . .
Remark. The simplest subsequence of Markov numbers (left branch of the above tree) contains
the odd Fibonacci numbers 1,2,5,13,34,89, . . .. It was noticed in [8], that the sequence
1,4,13,40,120,354,1031,2972,8495, . . .
appearing as the shadow of the odd Fibonacci branch, is the sequencel A238846 (see [13]) which
is the convolution of two bisections of the Fibonacci sequence, F2n+1 and F2n . We refer to [7, 8]
for various ε-deformations of Fibonacci numbers.
The combinatorial nature of other shadow sequence appearing in the above tree is
not understood. For instance, it would be interesting to know properties of the sequence
1,13,117,921,6761, . . . appearing as the shadow of 1,5,29,169,985, . . . which is a bisection of the
classical Pell numbers.
3. Mutation stability
In this section, we relax the positivity condition for the nilpotent part α of dual numbers A =
a +αε. We consider an arbitrary choice of the initial triplet (4) and give a slightly more general
form of the deformed Markov equation. We also obtain several properties of this equation.
3.1. Proof of Theorem 1
The following statement is a more precise and general reformulation of Theorem 1.
Proposition 2.
(i) Every triplet of dual numbers (A,B ,C ) obtained from (4) by a sequence of transforma-
tions (5) satisfies the equation
A2 +B 2 +C 2 = (3−σε) ABC , (8)
where σ=α1 +β1 +γ1.
(ii) Conversely, every triplet of solutions (A,B ,C ) to (8) with positive a,b,c can be obtained
from a triple of the form (4) by a sequence of transformations (5) mixed with permutations
of A,B and C .
Nathan Bonin and Valentin Ovsienko 1487
Proof. To prove Part (i), let us first check that the initial triplet (4) satisfies (8). Indeed, substitut-
ing (4) to the left-hand-side of the equation gives 3+2(α1+β1+γ1)ε, while in the right-hand-side
one obtains 3+3(α1 +β1 +γ1)ε−σε.
Let us check that (8) is stable with respect to the transformations (5).
Lemma 3. The equation
A2 +B 2 +C 2 = X ABC , (9)
where X is a formal parameter, is preserved by the transformations (5).
Proof of the lemma. Let us show that, if (A,B ,C ) is a solution to (9), then (A′,B ,C ), with A′ as
in (5), is also a solution to the same equation.
Assume that (A,B ,C ) is a solution to (9). Then A′ given by (5) can be rewritten:
A′ = X BC − A. (10)
Substituting this expression for A′ to the left-hand-side of (9), one obtains
A′2 +B 2 +C 2 = X 2 B 2C 2 −2X ABC + A2 +B 2 +C 2
= X 2 B 2C 2 −X ABC ,
while in the right-hand-side, one has
X A′BC = X (X BC − A)BC
= X 2 B 2C 2 −X ABC .
We conclude that the triplet (A′,B ,C ) satisfies exactly the same equation, namely A′2 +B 2 +C 2 =
X A′BC . □
The assumption that the initial triplet (A1,B1,C1) satisfies (9) implies that
X = A
2
1 +B 21 +C 21
A1B1C1
,
that gives X = 3− (α1 +β1 +γ1)ε.
Proposition 2(i) follows.
To prove Part (ii), we use the classical theorem that any positive solution (a,b,c) to the Markov
equation can be reduced to (1,1,1) by a sequence of transformations (3). We conclude following
the same sequence of transformations (5). □
3.2. Uniqueness
Let us show that (8) is the only ε-deformation of the Markov equation which is preserved by the
mutations (5).
Consider the most general ε-deformed Markov equation, it can be written in the form
A2 +B 2 +C 2 = 3ABC +P (A,B ,C )ε, (11)
where P is an arbitrary polynomial
P (A,B ,C ) = ∑
i , j ,k≥0
Pi j k A
i B j C k .
Theorem 4. Equation (8) is the only equation among the family (11) preserved by the transforma-
tions (5).
1488 Nathan Bonin and Valentin Ovsienko
Proof. Let us show that, if (11) is preserved by the transformations (5), then all the coefficients
Pi j k of the polynomial P vanish, except perhaps P111, which is arbitrary. Indeed, (5) can be
rewritten
A′ = 3BC − A+ P (A,B ,C )
A
ε
so that the left-hand-side of (11) after the transformation becomes
A′2 +B 2 +C 2 = 9B 2C 2 −6ABC + A2 +B 2 +C 2 +
(
6BC
A
P (A,B ,C )−2P (A,B ,C )
)
ε
= 9B 2C 2 −3ABC +
(
6BC
A
P (A,B ,C )−P (A,B ,C )
)
ε.
The right-hand-side of (11) is
3A′BC +P (A′,B ,C )ε= 9B 2C 2 −3ABC +
(
3BC
A
P (A,B ,C )+P (A′,B ,C )
)
ε.
The assumption the equation is preserved by the mutations then reads
P (A′,B ,C )ε=
(
3BC
A
−1
)
P (A,B ,C )ε,
or with more details ∑
i , j ,k≥0
Pi j k (3BC − A)i B j C k =
(
3BC
A
−1
) ∑
i , j ,k≥0
Pi j k A
i B j C k .
(Note that these terms appear with multiple ε, so that we omit P (A,B ,C )A ε in the left-hand-side
since ε2 = 0.)
The latter equation implies that Pi j k = 0 for i > 1, and similarly for j and k > 1. Indeed, taking
the maximal i , for which Pi j k ̸= 0, if i ≥ 2, all of the terms in the right-hand-side contain A,
while the left-hand-side contain the term Pi j k 3
i B i+ j C i+k . Therefore, the equation (11) cannot
be invariant in this case.
The theorem follows. □
Remark. Another version of super-Markov equation was suggested by Huang, Penner, and
Zeitlin [5]:
a2 +b2 + c2 + (ab +bc +ac)ε= 3(1+ε) abc.
For a discussion and a related conjecture; see [7]. This equation has interesting geometric
properties, albeit it is not preserved by the mutations (3).
4. Initial conditions
Let us end the note with a brief discussion of the choice of initial triplet α1,β1,γ1 in (4). The
choice of [8], which is (α1,β1,γ1) = (0,1,1) is not unique, but can be justified by the following.
A simple property of the equation (8) is that the shadow parts of its solutions linearly depend
on the choice of the initial triplet α1,β1,γ1. This readily follows from the second equation in (6).
Therefore, the shadow part of every solution to (8) is a linear combination of those with one of
the following initial triplets
(0,1,1), (1,1,1), (0,1,0),
that form a basis in the space of initial conditions. However, choosing (1,1,1), one obtains
the shadow part that coincides with the Markov numbers, namely the solutions of the form
(A,B ,C ) = (a + aε,b + bε,c + cε), which is not an interesting case. The choice (0,1,0) produces
negative integers. As mentioned in the introduction, it would be interesting to characterize the
initial triplets that produce only positive shadow parts.
Nathan Bonin and Valentin Ovsienko 1489
Acknowledgements
We would like to thank Philippe Larchevêque, Sophie Morier-Genoud, Julien Rouyer, Michael
Shapiro, Sergei Tabachnikov, Alexey Ustinov, and Alexander Veselov for fruitful discussions. We
are indebted to the anonymous referee for many helpful remarks and suggestions.
References
[1] C. H. Conley, V. Ovsienko, “Shadows of rationals and irrationals: supersymmetric continued fractions and the super
modular group”, 2022, https://arxiv.org/abs/2209.10426.
[2] S. Fomin, A. Zelevinsky, “Cluster algebras. I. Foundations”, J. Am. Math. Soc. 15 (2002), no. 2, p. 497-529.
[3] ——— , “The Laurent phenomenon”, Adv. Appl. Math. 28 (2002), no. 2, p. 119-144.
[4] A. N. W. Hone, “Casting light on shadow Somos sequences”, 2021, https://arxiv.org/abs/2111.10905.
[5] Y. Huang, R. C. Penner, A. M. Zeitlin, “Super McShane identity”, to appear in J. Differ. Geom., 2019, https://arxiv.org/
abs/1907.09978.
[6] A. Markov, “Sur les formes quadratiques binaires indéfinies”, Math. Ann. 15 (1879), p. 381-406.
[7] G. Musiker, N. Ovenhouse, S. W. Zhang, “Double Dimer Covers on Snake Graphs from Super Cluster Expansions”,
2021, https://arxiv.org/abs/2110.06497.
[8] V. Ovsienko, “Shadow sequences of integers, from Fibonacci to Markov and back”, to appear in Math. Intell., 2021,
https://arxiv.org/abs/2111.02553.
[9] V. Ovsienko, M. Shapiro, “Cluster algebras with Grassmann variables”, Electron. Res. Announc. Math. Sci. 26 (2019),
p. 1-15.
[10] V. Ovsienko, S. Tabachnikov, “Dual numbers, weighted quivers, and extended Somos and Gale-Robinson sequences”,
Algebr. Represent. Theory 21 (2018), no. 5, p. 1119-1132.
[11] J. Propp, “The combinatorics of frieze patterns and Markoff numbers”, Integers 20 (2020), article no. A12 (38 pages).
[12] J. M. Rabin, “Super elliptic curves”, J. Geom. Phys. 15 (1995), no. 3, p. 252-280.
[13] N. J. A. Sloane, “The On-Line Encyclopedia of Integer Sequences”, 2010, http://oeis.org.
[14] A. Veselov, “Conway’s light on the shadow of Mordell”, 2022, https://arxiv.org/abs/2208.14184.


A shadow Markov equation

Ballantine, Cristina

Merca, Mircea

Comptes Rendus
Mathématique
Cristina Ballantine and Mircea Merca
Congruences modulo 4 for the number of 3-regular partitions
Volume 361 (2023), p. 1577-1583
https://doi.org/10.5802/crmath.512
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1577-1583
https://doi.org/10.5802/crmath.512
Combinatorics, Number theory / Combinatoire, Théorie des nombres
Congruences modulo 4 for the number of
3-regular partitions
Cristina Ballantine ∗,a andMirceaMerca b,c
a Department of Mathematics and Computer Science, College of The Holy Cross, Worcester, MA 01610, USA
b Department of Mathematical Methods and Models, Fundamental Sciences Applied in Engineering
Research Center, University Politehnica of Bucharest, RO-060042 Bucharest, Romania
c Academy of Romanian Scientists, RO-050044, Bucharest, Romania
E-mails: cballant@holycross.edu, mircea.merca@profinfo.edu.ro
Abstract. The last decade has seen an abundance of congruences for bℓ(n), the number of ℓ-regular parti-
tions of n. Notably absent are congruences modulo 4 for b3(n). In this paper, we introduce Ramanujan type
congruences modulo 4 for b3(2n) involving some primes p congruent to 11,13,17,19,23 modulo 24.
Keywords. partitions, regular partitions, congruences.
2020 Mathematics Subject Classification. 11P83, 05A17, 11F33.
Manuscript received 14 March 2023, revised 19 April 2023, accepted 6 May 2023.
1. Introduction
A partition of a positive integer n is a weakly decreasing sequence of positive integers whose sum
is n. The positive integers in the sequence are called parts. For more on the theory of partitions,
we refer the reader to [1].
For an integer ℓ > 1, a partition is called ℓ-regular if none of its parts is divisible by ℓ. The
number of the ℓ-regular partitions of n is usually denoted by bℓ(n) and its arithmetic properties
were investigated extensively. See, for example, [3–7, 10–12, 17–20, 22]. The generating function
for bℓ(n) is given by ∞∑
n=0
bℓ(n) q
n = (q
ℓ; qℓ)∞
(q ; q)∞
.
Here and throughout q is a complex number with |q | < 1, and the symbol (a; q)∞ denotes the
infinite product
(a; q)∞ =
∞∏
n=0
(1−a qn).
In a recent paper, W. J. Keith and F. Zanello [9] discovered infinite families of Ramanujan type
congruences modulo 2 for b3(2n) involving every prime p with p ≡ 13,17,19,23 (mod 24).
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1578 Cristina Ballantine and Mircea Merca
Theorem 1 (Keith–Zanello). The sequence b3(2n) is lacunary modulo 2. If p ≡ 13,17,19,23
(mod 24) is prime, then
b3
(
2(p2n +pk −24−1))≡ 0 (mod 2)
for 1⩽ k ⩽ p −1, where 24−1 is taken modulo p2.
Motivated by the Keith–Zanello result, O. X. M. Yao [23] provided new infinite families of
Ramanujan type congruences modulo 2 for b3(2n) involving every prime p ⩾ 5.
Theorem 2 (Yao). Let p ⩾ 5 be a prime.
(1) If b3
(
p2−1
12
)
≡ 1 (mod 2), then for n,k ⩾ 0
b3
(
2p4k+4 n +2p4k+3 j + p
4k+4 −1
12
)
≡ 0 (mod 2)
where 1⩽ j ⩽ p −1 and for n,k ⩾ 0
b3
(
p4k −1
12
)
≡ 1 (mod 2).
(2) If b3
(
p2−1
12
)
≡ 0 (mod 2), then for n,k ⩾ 0 with p ∤ (24n +1)
b3
(
2p6k+2 n + p
6k+2 −1
12
)
≡ 0 (mod 2)
and for n,k ⩾ 0
b3
(
p6k −1
12
)
≡ 1 (mod 2).
Very recently, Ballantine, Merca and Radu [2] introduced new infinite Ramanujan type con-
gruences modulo 2 for b3(2n). They complement naturally the results of Keith–Zanello and Yao
and involve primes in the set
P = {p prime : ∃ j ∈ {1,4,8}, x, y ∈Z, gcd(x, y) = 1 with x2 +216y2 = j p}
whose Dirichlet density is 1/6.
Theorem 3. For every p ∈P and n ⩾ 0, we have
b3
(
2(p2 n +pα−24−1p )
)≡ 0 (mod 2),
where 0 ⩽α< p, α ̸= ⌊p/24⌋, and 24−1p is the inverse of 24 modulo p taken such that 1 ⩽−24−1p ⩽
p −1.
In this work, motivated by the results on the parity of b3(2n), we investigate Ramanujan type
congruences modulo 4 for b3(2n). We note that congruences modulo 4 for ℓ-regular partitions,
have been studied in [8] for ℓ = 4,5,9, and in [15] for ℓ = 2. However, congruences modulo 4 for
3-regular partitions are missing from the literature.
Theorem 4. For every p ∈ {43,47,59,61,67,89,137,139,157} and n ⩾ 0 we have
b3
(
2(p2 n +pα−24−1p )
)≡ 0 (mod 4), (1)
where 0⩽α< p, α ̸= ⌊p/24⌋, 24−1p is the inverse of 24 modulo p taken such that 1⩽−24−1p ⩽ p −1.
Cristina Ballantine and Mircea Merca 1579
We conjecture that there are infinitely many primes for which (1) holds. For example, we
verified numerically that, in addition to the primes in Theorem 4, the statement of the theorem
holds for
p ∈ {167,181,229,233,277,331,359,379,401,419,421,431,443,479,499,541,
569,593,599,613,643,647,691,709,719,757,761,787,809,827,829,853,
859,863,877,911,929,953,977,983,1021,1031}.
We were unable to finish the proof of Theorem 4 for these values of p due to computing time
limitations.
2. Proof of Theorem 4
2.1. Modular forms
As is the case with many proofs of congruences in the literature, we use [13, Lemma 4.5]. For
the conveniece of the reader, we first introduce all necessary notation and the statement of [13,
Lemma 4.5]. This exposition is nearly identical to that in [2].
Let Γ := SL(2,Z), and define
Γ∞ :=
{(
1 b
0 1
)
∈ Γ
}
.
For a positive integer N , we define the congruence subgroup
Γ0(N ) :=
{(
a b
c d
)
∈ Γ : c ≡ 0 (mod N )
}
.
If M is a positive integer, we write R(M) for the set of finite integer sequences r = (rδ1 ,rδ2 , . . . ,rδk ),
where 1 = δ1 < δ2 < ·· · < δk = M are the positive divisors of M . We note for the remainder of
this section we only consider positive divisors of a given integer. Given a positive integer m, we
denote by S24m the set of invertible quadratic residues modulo 24m and, for fixed 0 ⩽ t ⩽ m −1,
we define
Pm,r (t ) :=
{
t s + s −1
24
∑
δ|M
δrδ (mod m) : s ∈ S24m
}
.
Let m, M and N be positive integers. Moreover, let t be an integer such that 0⩽ t ⩽m −1 and
let r = (rδ) ∈ R(M). We set κ := gcd(1−m2,24) and write
∏
δ|M δ|rδ| =: 2s v , where s is a nonnegative
integer and v is odd. Then, we say that the tuple (n, M , N , (rδ), t ) ∈ ∆∗ if and only if all of the
following six conditions are satisfied.
(1) p | m, p prime, implies p | N ;
(2) δ | M , δ⩾ 1 such that rδ ̸= 0 implies δ | mN ;
(3) κN
∑
δ|M rδ mNδ ≡ 0 (mod 24);
(4) κN
∑
δ|M rδ ≡ 0 (mod 8);
(5) 24mgcd(κ(−24t−∑δ|M δrδ),24m)
∣∣∣N ;
(6) If 2 | m, then (4 | κN and 8 | N s) or (2 | s and 8 | N (1− v)).
Finally, for γ= (a bc d ) ∈ Γ, and m and r = (rδ) ∈ R(M) as above, we define
pm,r (γ) := min
d∈{0,...,m−1}
1
24
∑
δ|M
rδ
gcd2(δ(a +κdc),mc)
δm
and for a = (aδ) ∈ R(N ), we define
p∗a (γ) :=
1
24
∑
δ|N
aδ
gcd2(δ,c)
δ
.
1580 Cristina Ballantine and Mircea Merca
We use the notation ∞∑
n=0
c(n) qn ≡
∞∑
n=0
d(n) qn (mod u)
to mean that c(n) ≡ d(n) (mod u) for all n ⩾ 0. Similarly,
∞∑
n=0
c(n) qn ≡ 0 (mod u)
means c(n) ≡ 0 (mod u) for all n ⩾ 0.
Lemma 5 ([13, Lemma 4.5]). Let u be a positive integer, (m, M , N , t ,r = (rδ)) ∈ ∆∗, a = (aδ) ∈
R(N ). Let {γ1, . . . ,γn} ⊂ Γ be a complete set of representatives of the double cosets in Γ0(N )\Γ/Γ∞.
Assume that pm,r (γi )+p∗a (γi )⩾ 0 for all 0⩽ i ⩽ n. Let tmin := mint ′∈Pm,r (t ) t ′ and
ν := 1
24
((∑
δ|N
aδ+
∑
δ|M
rδ
)
[Γ : Γ0(N )]−
∑
δ|N
δaδ
)
− 1
24m
∑
δ|M
δrδ−
tmin
m
.
Suppose ∏
δ|M
∞∏
n=1
(1−qδn)rδ =
∞∑
n=0
cr (n)q
n .
If
⌊ν⌋∑
n=0
cr (mn + t ′)qn ≡ 0 (mod u), for all t ′ ∈ Pm,r (t ),
then ∞∑
n=0
cr (mn + t ′)qn ≡ 0 (mod u), for all t ′ ∈ Pm,r (t ).
2.2. Proof of Theorem 4
As customary, we use the notation
fi :=
∞∏
k=1
(1−qki ).
From [21], identity (2.18), we have
∞∑
n=0
b3(2n) q
n = f2 f3 f8 f
2
12
f 21 f4 f6 f24
. (2)
Moreover, since for i ≥ 1, f 22i ≡ f 4i (mod 4), we have
f2i
fi
= f
2
2i
f 4i
f 3i
f2i
≡ f
3
i
f2i
(mod 4). (3)
Using (2) and (3), we obtain
∞∑
n=0
b3(2n) q
n ≡ f
2
1 f3 f
3
4 f
3
6
f2 f8 f24
(mod 4).
To use the Lemma 5, we write
∞∑
n=0
c(n)qn := f
2
1 f3 f
3
4 f
3
6
f2 f8 f24
= ∏
δ|M
∞∏
n=1
(1−qδn)rδ .
Thus, with the notation of Lemma 5, we take u = 4, m = p2, M = 24, and
(r1,r2,r3,r4,r6,r8,r12,r24) = (2,−1,1,3,3,−1,0,−1).
Cristina Ballantine and Mircea Merca 1581
We have κ= 24 and we calculate∑
δ|M
rδ
δ
= 35
12
,
∑
δ|M
rδ = 6,
∑
δ|M
δrδ = 1.
We take N = 24p. It is straightforward to verify that for any t = pα−24−1p conditions (1)–(6) are
satisfied.
Since
[Γ : Γ0(N )] = N
∏
x|N
(1+x−1),
where the product is taken after all prime divisors of N , we have
[Γ : Γ0(N )] = 48(p +1).
In general, it is nontrivial to find a complete set of representatives for the double cosets
in Γ0(N ) \ Γ/Γ∞. If N is square free, it is shown in [14, Lemma 2.6] that a complete set of
representatives for Γ0(N )\Γ/Γ∞ is given by
AN =
{(
1 0
δ 1
)
: δ | N , δ⩾ 1
}
.
This result has been extended to N such that N /2 is square free in [16, Lemma 4.3]. While for
N = 24p, neither N nor N /2 is square free, when m = p2,κ= 24, and (r1,r2,r3,r4,r6,r8,r12,r24) =
(2,−1,1,3,3,−1,0,−1) we can avoid finding a complete set of representatives all together.
Since for any integers i , j ≥ 1 we have
gcd( j (a +κdc),mc)⩽ gcd(i j (a +κdc),mc)⩽ i gcd( j (a +κdc),mc),
an easy calculation shows that for each γ= (a bc d ) ∈ Γ, we have∑
δ|M
rδ
gcd2(δ(a +κdc),mc)
δm
≥ 0,
and thus pm,r (γ) ≥ 0. Hence, we can use a := (aδ)δ|N with aδ = 0 for each δ | N to calculate ⌊ν⌋. It
is clear from the definition of ν in Lemma 5 that
⌊ν⌋ = 12(p +1)−1.
Let
Rp := {pα−24−1p : 0⩽α< p, α ̸= ⌊p/24⌋}.
For each p, we used Mathematicatm to write the set Rp as
Rp = Pm,r (−24−1p )∪Pm,r (Ap −24−1p )
for a minimal A. For example, when p = 43, we have A = 2 and the Mathematicatm calculation
gives
Pm,r (−24−1p ) = {34,163,206,292,378,421,593,851,894,937,1023,1195,
1238,1281,1324,1367,1453,1496,1539,1668,1754}
and
Pm,r (2 ·43−24−1p ) = {120,249,335,464,507,550,636,679,722,765,808,980,
1066,1109,1152,1410,1582,1625,1711,1797,1840}.
If α= ⌊43/24⌋ = 1 we have Pm,r (p −24−1p ) = Pm,r (77) = {77}.
For p ∈ {43,47,139,157} we obtained A = 2 and for p ∈ {59,61,67,89,137} we obtained A = 1.
Moreover, if t∗ = p⌊p/24⌋−24−1p , then Pm,r (t∗) = {t∗}.
Finally, in each case, we verified that for each t ∈Rp we have
c(p2n + t ) ≡ 0 (mod 4) for 0⩽ n ⩽ 12(p +1)−1.
1582 Cristina Ballantine and Mircea Merca
Then, [13, Lemma 4.5] implies that, for each
p ∈U := {43,47,59,61,67,89,137,139,157}
we have
c(p2n + t ) ≡ 0 (mod 4) for all n ⩾ 0 and t ∈Rp .
Our calculations show that for each prime p ∈U , we have that
c(p⌊p/24⌋−24−1p ) ̸≡ 0 (mod 4)
and thus the requirement that α ̸= ⌊p/24⌋ in the statement of Theorem 4 is necessary.
3. Concluding remarks
Several Ramanujan type congruences modulo 4 for b3(2n) involving some primes p with
p ≡ 11,13,17,19,23 (mod 24) have been proved in this paper using modular forms and [13,
Lemma 4.5].
As mentioned in the introduction, we verified the statement of Theorem 4 numerically up to
108 for many more primes equivalent to 11,13,17,19,23 modulo 24. In fact, our computations
suggest that (1) is also true for primes p = 1033 and p = 1153 which are congruent to 1 modulo
24. We did not encounter any primes congruent to 5 or 7 modulo 24 for which (1) holds. We leave
it as an open problem to characterize an infinite family of primes for which the statement of
Theorem 4 holds and to prove the theorem for all these primes.
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[3] R. Carlson, J. J. Webb, “Infinite families of infinite families of congruences for k-regular partitions”, Ramanujan J. 33
(2014), no. 3, p. 329-337.
[4] S.-P. Cui, N. S. S. Gu, “Arithmetic properties of ℓ-regular partitions”, Adv. Appl. Math. 51 (2013), no. 4, p. 507-523.
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Cristina Ballantine and Mircea Merca 1583
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