QUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3

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1 QUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3 JEREMY LOVEJOY AND ROBERT OSBURN Abstract. Recenty, Andrews, Hirschhorn Seers have proven congruences moduo 3 for four types of partitions using eementary series manipuations. In this paper, we generaize their congruences 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. An overpartition of n is a partition of n where we may overine the first occurrence of a part. Let pn denote the number of overpartitions of n, p o n the number of overpartitions of n into odd parts, pedn the number of partitions of n without repeated even parts podn the number of partitions of n without repeated odd parts. The generating functions for these partitions are pnq n = q; q, 1.1 q; q p o nq n = q; q q; q, 1. pednq n = q ; q q; q, 1.3 podnq n = q; q q ; q, 1.4 where as usua a; q n := 1 a1 aq 1 aq n 1. The infinite products in are essentiay the four different ways one can speciaize the product aq; q /bq; q to obtain a moduar form whose eve is reativey prime to 3. A series of four recent papers examined congruence properties for these partition functions moduo 3 [1, 5, 6, 7]. Among the main theorems in these papers are the foowing congruences see Theorem 1.3 in [6], Coroary 3.3 Theorem 3.5 in [1], Theorem 1.1 in [5] Theorem 3. in [7], respectivey. For a n 0 α 0 we have p o 3 α An + B 0 mod 3, 1.5 where An + B = 9n + 6 or 7n + 9, Date: September, Mathematics Subject Cassification. Primary: 11P3; Secondary: 11E5. Key words phrases. partitions, overpartitions, congruences, binary quadratic forms, sums of squares. 1

2 JEREMY LOVEJOY AND ROBERT OSBURN ped 3 α+3 n α+ 1 ped 3 α+ n α mod 3, 1.6 p3 α 7n mod pod 3 α α mod We note that congruences moduo 3 for pn, p o n pedn are typicay vaid moduo 6 or 1. The powers of enter triviay or neary so, however, so we do not mention them here. The congruences in are proven in [1, 5, 6, 7] using eementary series manipuations. If we aow ourseves some eementary number theory, we find that much more is true. With our first resut we exhibit formuas for p o 3n ped3n + 1 moduo 3 for a n 0. These formuas depend on the factorization of n, which we write as n = a 3 b p v i i s q w j j, 1.9 where p i 1, 5, 7 or 11 mod 4 q j 13, 17, 19 or 3 mod 4. Further, et t denote the number of prime factors of n counting mutipicity that are congruent to 5 or 11 mod 4. Let Rn, Q denote the number of representations of n by the quadratic form Q. Theorem 1.1. For a n 0 we have p o 3n fnrn, x + 6y mod 3 ped3n n+1 Rn + 3, x + 3y mod 3, where fn is defined by { 1, n 1, 6, 9, 10 mod 1, fn = 1, otherwise. Moreover, we have p o 3n fn1 + 1 a+b+t 1 + v i s w j mod n ped3n + 1 p o 4n + 1 mod There are many ways to deduce congruences from Theorem 1.1. For exampe, cacuating the possibe residues of x + 6y moduo 9 we see that R3n +, x + 6y = R9n + 3, x + 6y = 0, then 1.10 impies that p o 7n p o 3n mod 3. This gives 1.5. The congruences in 1.6 foow from those in 1.5 after repacing 4n + 1 by 3 α 43n α 49n in We record two more coroaries, which aso foow readiy from Theorem 1.1.

3 QUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3 3 Coroary 1.. For a n 0 α 0 we have where An + B = 4n + 9 or 4n p o α An + B 0 mod 3, Coroary 1.3. If 1, 5, 7 or 11 mod 4 is prime, then for a n with n we have p o 3 n 0 mod For the functions p3n pod3n + we have reations not to binary quadratic forms but to r 5 n, the number of representations of n as the sum of five squares. Our second resut is the foowing. Theorem 1.4. For a n 0 we have p3n 1 n r 5 n mod 3 pod3n + 1 n r 5 n + 5 mod 3. Moreover, for a odd primes n 0, we have n p3 n + 1 p3n p3n/ mod where denotes the Legendre symbo. 1 n+1 pod3n + p4n + 15 mod 3, 1.14 Here we have taken p3n/ to be 0 uness 3n. Again there are many ways to deduce congruences. For exampe, 1.7 foows readiy upon combining 1.13 in the case = 3 with the fact that r 5 9n mod 3, which is a consequence of the fact that R9n + 6, x + y + 3z = 0. One can check that 1. foows simiary. For another exampe, we may appy 1.13 with n repaced by n for mod 3 to obtain Coroary 1.5. If mod 3 is prime n, then p3 3 n 0 mod 3.. Proofs of Theorems Proof of Theorem 1.1. On page 364 of [6] we find the identity where p o 3nq n = Dq3 Dq 6 Dq, Dq := n Z 1 n q n.

4 4 JEREMY LOVEJOY AND ROBERT OSBURN Reducing moduo 3, this impies that p o 3nq n 1 x+y q x +6y mod 3 x,y Z fnrn, x + 6y q n mod 3. Now it is known see Coroary 4. of [3], for exampe that if n has the factorization in 1.9, then Rn, x + 6y = a+b+t 1 + v i This gives Next, from [1] we find the identity where s w j..1 ped3n + 1q n = Dq3 ψ q 3 Dq, ψq := q nn+1/. Reducing moduo 3, repacing q by q mutipying by q 3 gives 1 n+1 ped3n + 1q n+3 Rn + 3, x + 3y q n+3 mod 3. It is known see Coroary 4.3 of [3], for exampe that if n has the factorization given in 1.9, then Rn, x + 3y = 1 1 a+b+t Comparing with.1 finishes the proof of v i Proof of Theorem 1.4. On page 3 of [5] we find the identity p3nq n Dq3 Dq Reducing moduo 3 repacing q by q yieds 1 n p3nq n s w j. mod 3. r 5 nq n mod 3. It is known see Lemma 1 in [4], for exampe that for any odd prime we have n r 5 n = r 5 n 3 r 5 n/.

5 QUADRATIC FORMS AND FOUR PARTITION FUNCTIONS MODULO 3 5 Here r 5 n/ = 0 uness n. Repacing r 5 n by 1 n p3n throughout gives Now equation 1 of [7] reads 1 n pod3n + q n = ψq3 3 ψq 4. Reducing moduo 3 we have 1 n pod3n + q n ψq 5 mod 3 r 5 n + 5q n mod 3 p4n + 15q n mod 3, where the second congruence foows from Theorem 1.1 in []. This impies 1.14 thus the proof of Theorem 1.4 is compete. Acknowedgement We woud ike to thank Scott Ahgren for pointing out reference [4]. The second author was partiay funded by Science Foundation Ire 0/RFP/MTH101. References [1] G. Andrews, M. Hirschhorn, J. Seers, Arithmetic properties of partitions with even parts distinct, Ramanujan J., to appear. [] P. Barruc, S. Cooper, M. Hirschhorn, Resuts of Hurwitz type for five or more squares, Ramanujan J. 6 00, [3] A. Berkovich H. Yesuyurt, Ramanujan s identities representation of integers by certain binary quaternary quadratic forms, Ramanujan J , [4] S. Cooper, Sums of five, seven, nine squares, Ramanujan J. 6 00, [5] M. Hirschhorn J. Seers, An infinite famiy of overpartition congruences moduo 1, INTEGERS 5 005, Artice A0. [6] M. Hirschhorn J. Seers, Arithmetic properties of overpartitions into odd parts, Ann. Comb , [7] M. Hirschhorn J. Seers, Arithmetic properties of partitions with odd parts distinct, Ramanujan J. 010, CNRS, LIAFA, Université Denis Diderot - Paris 7, Case 7014, 7505 Paris Cedex 13, FRANCE Schoo of Mathematica Sciences, University Coege Dubin, Befied, Dubin 4, Ire E-mai address: ovejoy@iafa.jussieu.fr E-mai address: robert.osburn@ucd.ie