QUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3
JEREMY LOVEJOY AND ROBERT OSBURN
Abstract. Recently, Andrews, Hirschhorn and Sellers have proven congruences modulo 3 for
four types of partitions using elementary series manipulations. In this paper, we generalize theircongruences using arithmetic properties of certain quadratic forms.

1. Introduction
A partition of a non-negative integer n is a non-increasing sequence whose sum is n. Anoverpartition of n is a partition of n where we may overline the first occurrence of a part. Letp(n) denote the number of overpartitions of n, po (n) the number of overpartitions of n into odd
parts, ped(n) the number of partitions of n without repeated even parts and pod(n) the numberof partitions of n without repeated odd parts. The generating functions for these partitions areX(−q; q)∞p(n)q n =,(1.1)
(q; q)∞n≥0

po (n)q n =(−q; q 2 )∞,(q; q 2 )∞(1.2)

ped(n)q n =(−q 2 ; q 2 )∞,(q; q 2 )∞(1.3)

pod(n)q n =(−q; q 2 )∞,(q 2 ; q 2 )∞(1.4)

Xn≥0Xn≥0X
n≥0where as usual(a; q)n := (1 − a)(1 − aq) · · · (1 − aq n−1 ).
The infinite products in (1.1)–(1.4) are essentially the four different ways one can specialize theproduct (−aq; q)∞ /(bq; q)∞ to obtain a modular form whose level is relatively prime to 3.
A series of four recent papers examined congruence properties for these partition functions
modulo 3 [1, 5, 6, 7]. Among the main theorems in these papers are the following congruences(see Theorem 1.3 in [6], Corollary 3.3 and Theorem 3.5 in [1], Theorem 1.1 in [5] and Theorem
3.2 in [7], respectively). For all n ≥ 0 and α ≥ 0 we havepo (32α (An + B)) ≡ 0
where An + B = 9n + 6 or 27n + 9,

(mod 3),(1.5)

Date: October 7, 2010.
2000 Mathematics Subject Classification. Primary: 11P83; Secondary: 11E25.
Key words and phrases. partitions, overpartitions, congruences, binary quadratic forms, sums of squares.
12

JEREMY LOVEJOY AND ROBERT OSBURN



2α+3ped 3
17 · 32α+2 − 1 
19 · 32α+1 − 1 
2α+2n+≡ ped 3n+≡088p(32α (27n + 18)) ≡ 0(mod 3),(mod 3)

(1.6)
(1.7)

and

23 · 32α+2 + 1
pod 3+≡ 0 (mod 3).
(1.8)
8We note that congruences modulo 3 for p(n), po (n) and ped(n) are typically valid modulo 6
or 12. The powers of 2 enter trivially (or nearly so), however, so we do not mention them here.
The congruences in (1.5)–(1.8) are proven in [1, 5, 6, 7] using elementary series manipulations.
If we allow ourselves some elementary number theory, we find that much more is true.
With our first result we exhibit formulas for po (3n) and ped(3n + 1) modulo 3 for all n ≥ 0.
These formulas depend on the factorization of n, which we write asrs
Y
Yw
vi
a b
n=2 3pi
qj j ,(1.9)


2α+3

i=1

j=1

where pi ≡ 1, 5, 7 or 11 (mod 24) and qj ≡ 13, 17, 19 or 23 (mod 24). Further, let t denote thenumber of prime factors of n (counting multiplicity) that are congruent to 5 or 11 (mod 24).
Let R(n, Q) denote the number of representations of n by the quadratic form Q.
Theorem 1.1. For all n ≥ 0 we havepo (3n) ≡ f (n)R(n, x2 + 6y 2 )(mod 3)andped(3n + 1) ≡ (−1)n+1 R(8n + 3, 2x2 + 3y 2 ) (mod 3),
where f (n) is defined by(−1, n ≡ 1, 6, 9, 10 (mod 12),f (n) =1,otherwise.
Moreover, we have
a+b+t

po (3n) ≡ f (n)(1 + (−1)

rsYY
1 + (−1)wj) (1 + vi )2i=1!
(mod 3)(1.10)j=1

and(−1)n ped(3n + 1) ≡ po (48n + 18)(mod 3).

(1.11)

There are many ways to deduce congruences from Theorem 1.1. For example, calculating thepossible residues of x2 + 6y 2 modulo 9 we see that
R(3n + 2, x2 + 6y 2 ) = R(9n + 3, x2 + 6y 2 ) = 0,and then (1.10) implies that po (27n) ≡ po (3n) (mod 3). This gives (1.5). The congruences in(1.6) follow from those in (1.5) after replacing 48n + 18 by 32α (48(3n + 2) + 18) and 32α (48(9n +6) + 18) in (1.11). We record two more corollaries, which also follow readily from Theorem 1.1.

QUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3

3Corollary 1.2. For all n ≥ 0 and α ≥ 0 we havepo (22α (An + B)) ≡ 0(mod 3),

where An + B = 24n + 9 or 24n + 15.
Corollary 1.3. If ` ≡ 1, 5, 7 or 11 (mod 24) is prime, then for all n with ` - n we havepo (3`2 n) ≡ 0(mod 3).

(1.12)For the functions p(3n) and pod(3n + 2) we have relations not to binary quadratic forms butto r5 (n), the number of representations of n as the sum of five squares. Our second result is thefollowing.
Theorem 1.4. For all n ≥ 0 we havep(3n) ≡ (−1)n r5 (n)(mod 3)

andpod(3n + 2) ≡ (−1)n r5 (8n + 5)

(mod 3).

Moreover, for all odd primes ` and n ≥ 0, we haven

p(3`2 n) ≡ ` − `+ 1 p(3n) − `p(3n/`2 ) (mod 3)`and(−1)n+1 pod(3n + 2) ≡ p(24n + 15) (mod 3),where •` denotes the Legendre symbol.

(1.13)(1.14)

Here we have taken p(3n/`2 ) to be 0 unless `2 | 3n. Again there are many ways to deduce
congruences. For example, (1.7) follows readily upon combining (1.13) in the case ` = 3 withthe fact thatr5 (9n + 6) ≡ 0 (mod 3),
which is a consequence of the fact that R(9n + 6, x2 + y 2 + 3z 2 ) = 0. One can check that (1.8)
follows similarly. For another example, we may apply (1.13) with n replaced by n` for ` ≡ 2(mod 3) to obtain
Corollary 1.5. If ` ≡ 2 (mod 3) is prime and ` - n, thenp(3`3 n) ≡ 0(mod 3).

2. Proofs of Theorems 1.1 and 1.4
Proof of Theorem 1.1. On page 364 of [6] we find the identity
Xpo (3n)q n =n≥0

D(q 3 )D(q 6 ),D(q)2

whereD(q) :=Xn∈Z

2(−1)n q n .

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JEREMY LOVEJOY AND ROBERT OSBURN

Reducing modulo 3, this implies thatXX
22
po (3n)q n ≡(−1)x+y q x +6yn≥0(mod 3)x,y∈Z≡Xf (n)R(n, x2 + 6y 2 )q n(mod 3).

n≥0

Now it is known (see Corollary 4.2 of [3], for example) that if n has the factorization in (1.9),
thenrsYY
1 + (−1)wj
2
2a+b+tR(n, x + 6y ) = (1 + (−1)) (1 + vi )
2i=1!
.

(2.1)

j=1

This gives (1.10). Next, from [1] we find the identity
Xped(3n + 1)q n =n≥0

D(q 3 )ψ(−q 3 )
,
D(q)2

where
ψ(q) :=Xq n(n+1)/2 .

n≥0Reducing modulo 3, replacing q by −q 8 and multiplying by q 3 gives
X(−1)n+1 ped(3n + 1)q 8n+3 ≡n≥0X

R(8n + 3, 2x2 + 3y 2 )q 8n+3(mod 3).

n≥0

It is known (see Corollary 4.3 of [3], for example) that if n has the factorization given in (1.9),
thenrsYY
1 + (−1)wj
R(n, 2x2 + 3y 2 ) = (1 − (−1)a+b+t ) (1 + vi )
2i=1!
.

j=1

Comparing with (2.1) finishes the proof of (1.11).

Proof of Theorem 1.4. On page 3 of [5] we find the identity
Xn≥0p(3n)q n ≡D(q 3 )2D(q)(mod 3).

Reducing modulo 3 and replacing q by −q yields
XX(−1)n p(3n)q n ≡r5 (n)q nn≥0(mod 3).

n≥0

It is known (see Lemma 1 in [4], for example) that for any odd prime ` we haven
r5 (`2 n) = `3 − `+ 1 r5 (n) − `3 r5 (n/`2 ).
`QUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 35Here r5 (n/`2 ) = 0 unless `2 | n. Replacing r5 (n) by (−1)n p(3n) throughout gives (1.13). Nowequation (1) of [7] readsXψ(q 3 )3.
(−1)n pod(3n + 2)q n =
ψ(q)4
n≥0

Reducing modulo 3 we haveX(−1)n pod(3n + 2)q n ≡ ψ(q)5(mod 3)n≥0≡

Xr5 (8n + 5)q n(mod 3)n≥0≡ −Xp(24n + 15)q n(mod 3),n≥0where the second congruence follows from Theorem 1.1 in [2]. This implies (1.14) and thus theproof of Theorem 1.4 is complete.
Acknowledgement
We would like to thank Scott Ahlgren for pointing out reference [4]. The second author waspartially funded by Science Foundation Ireland 08/RFP/MTH1081.
References[1] G. Andrews, M. Hirschhorn, and J. Sellers, Arithmetic properties of partitions with even parts distinct,Ramanujan J., to appear.
[2] P. Barrucand, S. Cooper, and M. Hirschhorn, Results of Hurwitz type for five or more squares, Ramanujan
J. 6 (2002), 347–367.
[3] A. Berkovich and H. Yesulyurt, Ramanujan’s identities and representation of integers by certain binary andquaternary quadratic forms, Ramanujan J. 20 (2009), 375–408.
[4] S. Cooper, Sums of five, seven, and nine squares, Ramanujan J. 6 (2002), 469–490.
[5] M. Hirschhorn and J. Sellers, An infinite family of overpartition congruences modulo 12, INTEGERS 5(2005), Article A20.
[6] M. Hirschhorn and J. Sellers, Arithmetic properties of overpartitions into odd parts, Ann. Comb. 10 (2006),
353–367.
[7] M. Hirschhorn and J. Sellers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 22(2010), 273–284.
CNRS, LIAFA, Université Denis Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, FRANCE
School of Mathematical Sciences, University College Dublin, Belfield, Dublin 4, IrelandE-mail address: lovejoy@liafa.jussieu.fr
E-mail address: robert.osburn@ucd.ie