CONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE POD FUNCTION

Transcription

1 CONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE POD FUNCTION SILVIU RADU AND JAMES A. SELLERS Abstract. In this aer, we rove arithmetic roerties modulo 5 7 satisfied by the function odn which denotes the number of artitions of n wherein odd arts must be distinct even arts are unrestricted. In articular, we rove the following: For all n 0, od135n mod 5, od135n mod 5, od135n mod 5, od675n mod 25, od3375n mod 125, od3375n mod 125, od567n mod 7, od567n mod Introduction The focus of this aer is the function odn which denotes the number of artitions of n in which odd arts are distinct even arts are unrestricted. This function odn has been considered by many from a roduct series oint of view as well as from other directions. For examle, odn aears in the works of Andrews [2, 3] Berkovich Garvan [6]. Moreover, Berkovich Garvan note that Andrews [5] considered a restricted version of odn wherein each art was required to be larger than 1. In very recent work, Alladi [1] obtained a series exansion for the roduct generating function for odn. It is significant to note that Hirschhorn Sellers [7] aear to be the first to consider odn from an arithmetic viewoint. In contrast to the work of Hirschhorn Sellers [7], in which odn was extensively studied modulo 3, we now wish to rove Ramanujan like roerties modulo 5 7 which are satisfied by odn. In articular, we rove the following theorem: Date: May 9, Mathematics Subject Classification. 05A17, 11P83. Key words hrases. congruences, modular forms, artitions, od function. S. Radu was suorted by DK grant W1214-DK6 of the Austrian Science Funds FWF. J. A. Sellers gratefully acknowledges the leadershi of the Research Institute for Symbolic Comutation RISC, Austria, for suorting his visit to the Institute in May 2010 when this research was initiated. 1

2 2 S. RADU AND J. A. SELLERS Theorem 1.1. For all n 0, 1 od135n + 8 od135n od135n mod 5, 2 od675n mod 25, 3 od3375n od3375n mod 125, 4 od567n od567n mod 7. For the roof of our congruences we need the following lemma. Lemma 1.2. Let be a rime α a ositive integer. Then 1 q n α 5 1 mod α. 1 q n α 1 Proof. We note that for all rimes X an indeterminate we have 6 X 1 mod α X 1 mod α+1. We see that 5 is true for α 1 because of the relation 1 q n 1 q n mod. Next we rove that if 5 is true for α N with N 1, then 5 is true for α N + 1. This follows by alying 6 with X 1 q n N 1 q n N 1. By elementary artition theory we see that 7 odmq m m0 1 + q 2n 1 1 q 2n. From here, we can rove some additional elementary generating function results which are critical to our roof of these congruences. Lemma 1.3. odm q m 1 q n 1 q 2n 2. m0 Proof. By 7 we find m0 odm q m 1 q 2n 1 1 q 2n 1 q n 1 q 2n 1 1 q 2n 1 q n 1 q 2n 2.

3 CONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE POD FUNCTION 3 In order to rove the congruences 1-4 we could use Lemma 2.4 below directly. However, exeriments show that a simle re rocessing of the congruences before the alication of Lemma 2.4 gives us a roof where fewer comutations are required. For this urose we use the following related generating function lemma to rewrite 1-4 in a form more convenient for us. Definition 1.4. For all ositive integers α rimes we define od α, mq m 1 q n α +1 :. 1 q 2n 2 1 q n α 1 m0 Lemma 1.5. The congruences 1-4 are true iff, for all n 0, od 1,5 135n + 8 od 1,5 135n od 1,5 135n mod 5, od 2,5 675n mod 25, od 3,5 3375n od 3,5 3375n mod 125, od 1,7 567n od 1,7 567n mod 7. Proof. The lemma follows immediately by observing that od α, n 1 n odn mod α, which follows from Lemma 1.2, Lemma 1.3 Definition The Main Proof Machinery Modular Forms For M a ositive integer let RM be the set of integer sequences indexed by the ositive divisors δ of M. Let 1 δ 1 < < δ k M be the ositive divisors of M r RM. Then we will write r r δ1,...,r δk. For s an integer m a ositive integer we denote by [s] m the set of all elements congruent to s modulo m, in other words [s] m Z m. Let Z m be the set of all invertible elements in Z m. Let S m Z m be the set of all squares in Z m. Definition 2.1. For m,m N, r r δ RM t {0,...,m 1} we define the ma r : S 24m {0,...,m 1} {0,...,m 1} with [s] 24m,t [s] 24m r t the image is uniquely determined by the relation [s] 24m r t ts+ s 1 24 δ M δr δ mod m. We define the set Let a Z an odd rime, then P m,r t : {[s] 24m r t [s] 24m S 24m }. a is the Legendre symbol. Lemma 2.2. Let 5 be a rime α a ositive integer. Let r α, : r α, 1,r α, 2,r α, 1 + α, 2, α 1 R2. Let a,b be ositive integers, m : 3 a b g : gcdm,8t 1. Then if 3 a 1 b 1 8t 1

4 4 S. RADU AND J. A. SELLERS we have P m,r α,t { t g 8t 1, 8t 1/g 8t 1/g 0 t m 1 for each m g, }. Proof. By Definition 2.1 we have P m,r α,t {t t ts + s 1 δr α, δ mod m,0 t m 1,[s] 24m S 24m } 24 δ m {t t ts + 1 s mod m,0 t m 1,[s] 24m S 24m } 8 {t s8t 1 8t 1 mod m,0 t m 1,[s] 24m S 24m } { } t g 8t 1,s8t 1/g 8t 1/g mod m/g, 0 t. m 1,[s] m/g S m/g The roof is finished by noting that the existence of [s] m/g S m/g such that is, for the case m g s8t 1/g 8t 1/g mod m/g squarefree, equivalent to 8t 1/g 8t 1/g, for each m g. We also used the fact that the canonical homomorhism φ : S n S n/d is surjective for any ositive integers n,d such that d n. We now use Lemma 2.2 to comute P m,r α,t for α,,m,t 1,5,135,8,1,5,135,107,2,5,675,647,3,5,3375,1997, 1,7,567,260 1,7,567,449. α,,m,t 1,5,135,8 : We see that g gcd135, t 1/g is for for 3. By Lemma 2.2 we need to solve the following equations for t : 8t 1/g t 1/g We see that x 5 1 has the solutions x 2,3 mod 5 x 3 1 has the solution x 1 mod 3. By the Chinese Remainder Theorem we obtain x 7,13 mod 15. Consequently we need to solve the following congruences for t : which is equivalent to hence 8t 1/g 7,13 mod 15, 8t 1 7g,13g mod 15g, t 1 + 7g/8,13g + 1/8 mod 15g. Finally using g 9 we obtain t 8,116 mod 135. This shows that 8 P 135,r 1,58 {8,116}.

5 CONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE POD FUNCTION 5 α,,m,t 1,5,135,107 : We see that g gcd135, t 1/g Note that the only rime which divides m/g 3 is 3. By Lemma 2.2 we need to solve the following equation for t : 8t 1/g This gives 8t 1/g 1 mod 3 8t 1 g mod 3g t 1 + g/8 mod 3g. Using g 45 we obtain t 107 mod 135. We conclude 9 P 135,r 1,5107 {107}. Alying Lemma 2.2 in analogous fashion we obtain: P 675,r 2,5647 {647}, P 3375,r 3,51997 {1997,3347}, P 567,r 1,7260 {260}, P 567,r 1,7449 {449}. By using 8-13 Lemma 1.5 we see that Theorem 1.1 can be rewritten as: Lemma 2.3. The congruences in Theorem 1.1 are true iff, for all n 0, od 1,5 135n + t 0 mod 5, t P 135,r 1,58, od 1,5 135n + t 0 mod 5, t P 135,r 1,5107, od 2,5 675n + t 0 mod 25, t P 675,r 2,5647, od 3,5 3375n + t 0 mod 125, t P 3375,r 3,51997, od 1,7 567n + t 0 mod 7, t P 567,r 1,7260, 19 od 1,7 567n + t 0 mod 7, t P 567,r 1,7449. For each r RM we assign a generating function f r q : 1 q δn r δ c r nq n. δ M n0 Given a rime, m N t {0,...,m 1} we are concerned with roving congruences of the tye c r mn + t 0 mod,n N. The congruences we are concerned with here have some additional structure; namely c r mn + t 0 mod,n 0,t P m,r t. In other words a congruence is a tule r,m,m,t, with r RM, m 1, t {0,...,m 1} a rime such that c r mn + t 0 mod,n 0,t P m,r t. Throughout when we say that c r mn+t 0 mod we mean that c r mn+t 0 mod for all n 0 all t Pt. In order to rove the congruences 1-4 we need a lemma [8, Lemma 4.5]. We first state it then exlain the terminology.

6 6 S. RADU AND J. A. SELLERS Lemma 2.4. Let u be a ositive integer, m,m,n,t,r r δ, a a δ RN, n the number of double cosets in Γ 0 N\Γ/Γ {γ 1,...,γ n } Γ a comlete set of reresentatives of the double coset Γ 0 N\Γ/Γ. Assume that m,r γ i + aγ i 0, i {1,...,n}. Let t min : min t P m,rt t ν : 1 a δ + r δ [Γ : Γ 0 N] δa δ 1 δr δ t min 24 24m m. δ N δ M δ N Then if for all t P m,r t then for all t P m,r t. ν c r mn + t q n 0 mod u n0 c r mn + t q n 0 mod u n0 δ M The lemma reduces the roof of a congruence modulo u to checking that finitely many values are divisible by u. We first define the set. Let κ κm gcdm 2 1,24 πm,r δ : s,j where s is a non-negative integer j an odd integer uniquely determined by δ M δ r δ 2 s j. Then a tule m,m,n,r δ,t belongs to iff m,m,n are ositive integers, r δ RM, t {0,...,m 1}; m imlies N for every rime ; δ M imlies δ mn for every δ 1 such that r δ 0; κn δ M r δ mn δ 0 mod 24; κn δ M r δ 0 mod 8; 24m gcdκ 24t δ M δr δ,24m N; for s,j πm,r δ we have 4 κn 8 Ns or 2 s 8 N1 j if 2 m. Remark 2.5. We note that the condition 2 m in the last line is not in the definition of in [8]. However every result in [8] holds with no modification having this extra condition. In fact this condition was somehow missed in [8] when was defined although the results hold without it, in some cases we obtain less otimality. Next we need to define the grous Γ,Γ 0 N Γ : { } a b Γ : a,b,c,d Z,ad bc 1, c d { } a b Γ 0 N : Γ N c c d for N a ositive integer, { } 1 h Γ : h Z. 0 1

7 CONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE POD FUNCTION 7 For the index we have 20 [Γ : Γ 0 N] : N N1 + 1 see, for examle, [9]. a b Finally for m, M, N ositive integers, r RM, a RN γ c d define 1 gcd 2 δa + κλc,mc m,r γ : min r δ λ {0,...,m 1} 24 δm aγ : 1 24 δ M δ M a δ gcd 2 δ,c. δ we Lemma 2.6. Let N be a squarefree integer. Then 1 0 Γ 0 N Γ Γ. δ 1 δ N Proof. Let a b c d Γ. Then if h Z such that 21 c + ch dgcdc,n 0 mod N we have a c b d 1 0 This imlies that h Γ 0 N. gcdc,n 1 c + ch dgcdc,n a b c d Γ 0 N In articular we have roven that γ a c γ δ N Γ 0 N Γ. 1 0 gcdc,n 1 Γ imlies b d 1 δ 0 1 Γ, if for any c,d Z with gcdc,d 1 there exists a h Z such that 21 holds. Next observe that 21 is equivalent to ch d c gcdc,n mod N/gcdc,N, which has a solution if gcdc,n/gcdc,n 1. This is always true because N is squarefree.

8 8 S. RADU AND J. A. SELLERS 3. The Congruences We start by roving 14. We aly Lemma 2.4 with m,m,n,t,r r 1,r 2,r 5 135,10,30,8,6, 2, 1, a a 1,a 2,a 3,a 5,a 6,a 10,a 15,a 30 7,14,2,0, 4,0,0,0. For δ Z let γ δ : 1 0 δ 1. Then by Lemma 2.6 a comlete set of double coset reresentatives is contained in the set Hence verifying the condition {γ δ : δ N}. m,r γ δ + aγ δ 0 for each δ N is sufficient to fulfill the assumtion of Lemma 2.4. This verification has been carried out using MAPLE. Next we obtain ν : 1 a δ + r δ [Γ : Γ 0 N] δa δ 1 δr δ t min 24 24m m δ N δ M δ N δ M Here we have used 20 to comute [Γ : Γ 0 30] This gives ν 23. By Lemma 2.4 we obtain that 22 od 1,5 135n + 8 od 1,5 135n mod 5 for each 0 n 23 imlies od 1,5 135n + 8 od 1,5 135n mod 5 for all n 0. We have verified 22 with MAPLE. This roves 14. In an analogous fashion, alying Lemma 2.4 we rove the congruences below: Congruence 15. We aly Lemma 2.4 with m,m,n,t,r r 1,r 2,r 5 135,10,30,107,6, 2, 1 a a 1,a 2,a 3,a 5,a 6,a 10,a 15,a 30 7,14,2,0, 4,0,0,0. We obtain ν 23 Pt {107}. Congruence 16. We aly Lemma 2.4 with m,m,n,t,r r 1,r 2,r 5 675,10,30,647,26, 2, 5 a a 1,a 2,a 3,a 5,a 6,a 10,a 15,a 30 32,64,10,6, 20, 12, 2, 4. We obtain ν 109 Pt {647}. Congruence 17. We aly Lemma 2.4 with m,m,n,t,r r 1,r 2,r ,10,30,1997,126, 2, 25

9 CONGRUENCE PROPERTIES MODULO 5 AND 7 FOR THE POD FUNCTION 9 a a 1,a 2,a 3,a 5,a 6,a 10,a 15,a ,317,54,32, 106, 62, 11,21. We obtain ν 554 Pt {1997, 3347}. Congruence 18. We aly Lemma 2.4 with m,m,n,t,r r 1,r 2,r 7 567,14,42,260,8, 2, 1 a a 1,a 2,a 3,a 7,a 6,a 14,a 21,a 42 13,26,4, 8,0,0,0,0. We obtain ν 55 Pt {260}. Congruence 19. We aly Lemma 2.4 with m,m,n,t,r r 1,r 2,r 7 567,14,42,449,8, 2, 1 a a 1,a 2,a 3,a 7,a 6,a 14,a 21,a 42 13,26,4, 8,0,0,0,0. We obtain ν 55 Pt {449}. The above information is summarized in the following table: Cong. m, M, N, t, r ν a Pt , 10, 30, 8, 6, 2, , 14, 2, 0, 4, 0, 0, 0 {8, 116} , 10, 30, 107, 6, 2, , 14, 2, 0, 4, 0, 0, 0 {107} , 10, 30, 647, 26, 2, , 64, 10, 6, 20, 12, 2, 4 {647} , 10, 30, 1997, 126, 2, , 317, 54, 32, 106, 62, 11, 21 {1997, 3347} , 14, 42, 260, 8, 2, , 26, 4, 8, 0, 0, 0, 0 {260} , 14, 42, 449, 8, 2, , 26, 4, 8, 0, 0, 0, 0 {449} In each of the cases, we used MAPLE to verify that the congruences hold u to the bound ν. Thus, by Lemma 2.4 we have roven Hence, by Lemma 1.5, we have roven Theorem Notes On Comutations In our roofs above, we needed to check the divisibility by α of od α, n for certain α,n N a rime. However, we observe that α od α, n α odn α 1 n odn. These facts simlify the check of divisibility because we can build a nice recurrence for 1 n odn which we deduce in the following way. From Jacobi s Trile Product Identity [4, Theorem 2.8], we see that 1 + q n2n 1 + q n2n+1 q n2n+1 1 q 2n 2 1 q n. n Z Together with Lemma 1.3, we have 1 + q n2n 1 + q n2n+1 1 n odnq n 1. n0

10 10 S. RADU AND J. A. SELLERS Therefore, by using the formula for the Cauchy roduct of two sequences simlifying, one obtains the following for all ositive integers n: 1 n+1 odn odn k2k 1 1 n k + k 1,k2k 1 n k 1,k2k+1 n odn k2k n k. This rovides an extremely efficient method for calculating the values of odn which are needed to comlete our roofs. 5. Closing Thoughts It is truly satisfying to rove these congruences modulo 5 7 for the od function. However, our ultimate goal was to identify an infinite family of congruences modulo arbitrarily large owers of 5 or 7 satisfied by odn. Unfortunately, we were unable to find such a family. We may return to this theme in the future. References [1] K. Alladi. Partitions with non-reeating odd arts q hyergeometric identities. to aear. [2] G. E. Andrews. A generalization of the göllnitz gordon artition theorems. Proc. Amer. Math. Soc., 8: , [3] G. E. Andrews. Two theorems of gauss allied identities roved arithmetically. Pac. J. Math., 41: , [4] G. E. Andrews. The Theory of Partitions. Addison-Wesley, [5] G. E. Andrews. Partitions durfee dissection. Amer. J. Math., 101: , [6] A. Berkovich F. G. Garvan. Some observations on dyson s new symmetries of artitions. J. Comb. Thy. Ser. A, 1001:61 93, [7] M. D. Hirschhorn J. A. Sellers. Arithmetic roerties of artitions with odd arts distinct. Ramanujan Journal, to aear. [8] S. Radu. An Algorithmic Aroach to Ramanujan s Congruences. Ramanujan Journal, 202: , [9] G. Shimura. Introduction to the Arithmetic Theory of Automorhic Functions. Princeton University Press, Research Institute for Symbolic Comutation RISC, Johannes Keler University, A Linz, Austria, sradu@risc.uni-linz.ac.at Deartment of Mathematics, Penn State University, University Park, PA 16802, USA, sellersj@math.su.edu