Congruences of Multipartition Functions Modulo Powers of Primes

Transcription

1 Congruences of Multipartition Functions Modulo Powers of Primes William YC Chen 1, Daniel K Du, Qing-Hu Hou 3 and Lisa HSun 4 Center for Combinatorics, LPMC-TJKLC Nankai University, Tianjin , P R China 1 chen@nankaieducn, dukang@mailnankaieducn, 3 hou@nankaieducn, 4 sunhui@nankaieducn Abstract Let p r n denote the number of r-component multipartitions of n, and let S γ,λ be the space spanned by η4z γ φ4z, where ηz is the Dedekind s eta function and φz is a holomorphic modular form in M λ SL Z Inthispaper,weshowthatthegeneratingfunctionofp r mk n+r 4 with respect to n is congruent to a function in the space S γ,λ modulo m k As special cases, this relation leads to many well known congruences including the Ramanujan congruences of pn modulo 5, 7, 11 and Gandhi s congruences of p n modulo 5 and p 8 n modulo 11 Furthermore, using the invariance property of S γ,λ under the Hecke operator T l, we obtain two classes of congruences pertaining to the m k -adic property of p r n AMS Classification 05A17, 11F33, 11P83 Keywords modular form, partition, multipartition, Ramanujan-type congruence 1 Introduction The objective of this paper is to use the theory of modular forms to derive certain congruences of multipartitions modulo powers of primes Recall that an ordinary partition λ of a nonnegative integer n is a nonincreasing sequence of positive integers whose sum is n, where n is called the weight of λ The partition function pn is defined to be the number of partitions of n A multipartition of n with r components, as called by Andrews [], also referred to as an r-colored partition, see, for example [7, 9], is an r-tuple λ = λ 1,,λ r of partitions whose weights sum to n The number of r-component multipartitions of n is denoted by p r n Multipartitions arise in combinatorics, representation theory and physics As pointed out by Fayers [10], the representations of the Ariki-Koike algebra are naturally indexed by multiparititions Bouwknegt [6] showed that the Durfee square formulas of multipartitions are useful in deriving expressions for the characters of modules of affine Lie algebras in terms of the universal chiral partition functions 1

2 For the partition function pn, Ramanujan [3 6] proved that pan+b 0 mod M, 11 for all nonnegative integers n and for A, B, M = 5, 4, 5, 7, 5, 7 and 11, 6, 11 In general, congruences of form 11 are called Ramanujan-type congruences For m = 5 and 7, Watson [9] proved that pm k n+β m,k 0 mod m k, 1 where k 1 and β m,k 1/4 mod m k Atkin [3] showed that 1 is also valid for m = 11 When M is not a power of 5,7 or 11, Atkin and O Brien [5] discovered the following congruence p n+37 0 mod 13 Using thetheoryofmodularforms, Ono[1]proved thatforanyprimem 5 and positive integer k, there is a positive proportion of primes l such that m k l 3 n+1 p 0 mod m 13 4 holds for every nonnegative integer n coprime to l Weaver [30] gave an algorithm for finding the values of l in 13 for primes 13 m 31 Ramanujan-type congruences of p r n have been extensively studied, see for example [, 4, 11, 1, 14, 16, 19, 8] Gandhi [11] derived the following congruences of p r n by applying the identities of Euler and Jacobi p 5n+3 0 mod 5, 14 p 8 11n+4 0 mod With the aid of Sturm s theorem [7], Eichhorn and Ono [9] computed an upper bound CA,B,r,m k such that p r An+B 0 mod m k holdsforallnonnegativeintegersnifandonlyifitistrueforn CA,B,r,m k For example, to prove 14, it suffices to check that it holds for n 3 In the same vain, one can prove 15 by verifying that it holds for n 11 Treneer [8] extended 13 to weakly holomorphic modular forms and showed that for any prime m 5 and positive integers k, there is a positive proportion of primes l such that p r m k l μr n+r 4 0 mod m

3 for every nonnegative integer n coprime to l, where μ r equals to 1 if r is even and 3 if r is odd The aim of this paper is to study congruence properties of p r n modulo powers of primes For example, we shall show that m k l μk 1 n+r p r 0 mod m k, 16 4 where r is an odd integer, l is a prime other than,3 and m, and μ is a positive integer, K is a fixed positive integer, and n is a positive integer coprime to l To derive congruences of p r n, one may consider the congruence properties of the generating functions of p r n For the case of ordinary partitions, ie, r = 1, Chua [8] showed that mn+1 p q n η4z γm φ m 4z mod m, 17 4 mn 1 mod 4 where ηz is Dedekind s eta function, γ m is an integer depending on m and φ m z isaholomorphicmodular form Ahlgren andboylan[1] extended 17 to congruences modulo powers of primes, namely, m k n+1 F m,k z = p q n η4z γ m,k φ m,k 4z mod m k, 4 m k n 1 mod 4 where γ m,k is an integer and φ m,k z is a holomorphic modular form m= n r m p r n+r 4 18 In order to prove the existence of congruences of p r n modulo powers of primes, Brown and Li [7] introduced the generating function G m,k,r z q n mod m k, 19 and showed that G m,k,r z is a modular form of level 576m 3 Kilbourn [15] used the generating function mn+r H m,k,r z p r q n mod m k, mn r mod 4 and proved that H m,k,r z is a modular form of level 576m However, due to the large dimensions of the spaces M λ Γ 0 576m 3 and M λ Γ 0 576m, it does not seem to be a feasible task to compute explicit bases In other words, to derive explicit congruence formulas of p r n, it is desirable to find 3

4 a generating function of p r n that can be expressed in terms of modular forms of a small level In this paper, we find the following extension of the generating function F m,k z, namely, F m,k,r z = m k n r mod 4 m k n+r p r q n, where q = e πiz We show that F m,k,r z is congruent to a meromorphic function modulo m k More precisely, we find F m,k,r z η4z γ m,k,r φ m,k,r 4z mod m k, 11 where γ m,k,r is an integer and φ m,k,r z is a holomorphic modular form in M λm,k,r SL Z Noting that any element of M λm,k,r SL Z can be expressed as a polynomial of the Eisenstein series E 4 z and E 6 z This enable us to derive explicit congruences of generating functions of p r n modulo m k If φ m,k,r z = 0, then 11 yields a Ramanujan-type congruence as follows m k n+r p r 0 mod m k For example, it is easily checked that φ 5,1, z = 0 and φ 11,1, z = 0, hence Gandhi s congruences 14 and 15 are the consequences of 113 We also find p 5 n+3 0 mod 5, 114 p 8 11 n+81 0 mod 11, 115 since φ 5,, z = 0 and φ 11,,8 z = 0 For more congruences of form 113, see Table 5 On the other hand, if φ m,k,r z = 0 in 11, we may use Yang s method [31] to find congruences of form 16 For example, since F 5,,3 z is congruent to a modular form in the invariant space S 1,48 of T 5 modulo 5, we have Preliminaries p n mod 5 To make this paper self-contained, we recall some definitions and facts on modular forms In particular, we shall use the U-operator, the V-operator, the Hecke operator and the twist operator on the modular forms 4

5 Let k 1 Z be an integer or a half-integer, N be a positive integer with 4 N if k Z and χ be a Nebentypus character We use M k Γ 0 N,χ to denote the space of holomorphic modular forms on Γ 0 N of weight k and character χ The corresponding space of cusp forms is denoted by S k Γ 0 N,χ If χ is the trivial character, we shall write M k Γ 0 N and S k Γ 0 N for M k Γ 0 N,χ and S k Γ 0 N,χ Moreover, we write SL Z for Γ 0 1 Let fz M k Γ 0 N,χ with the following Fourier expansion at fz = n 0anq n, where q = e πiz Let us recall some operators acting on fz Let γ = a b c d be a real matrix with positive determinant The k slash operator k is defined by f k γz = detγ k/ cz +d k fγz, 1 where γz = az +b cz +d In particular, let l be an integer and 0 1 γ l = l 0 The Fricke involution W l is given by f W l = f k γ l The U-operator U l and V-operator V l are defined by fz U l = n 0alnq n, 3 and fz V l = n 0anq ln 4 It is known that l 1 fz k U l = l k 1 1 μ fz k 0 l μ=0 5 5

6 Let ψ be a Dirichlet character The ψ-twist of fz is defined by f ψz = n 0ψnanq n Let l be a prime and fz M λ+ 1Γ 0 N,χ be a modular form of halfintegral weight The Hecke operator T l is defined by 1 λ n fz T l = n 0 al n+χl l l λ 1 an+χl l λ 1 a n l q n 6 We will use the following level reduction properties of the operators U l and F l = U l +l k 1 W l see [17, Lemma 1] and [8, Lemma ] Lemma 1 Let k Z, N be a positive integer, χ be a character modulo N, and fz M k Γ 0 N,χ Assume that l is a prime factor of N and χ is also a character modulo N/l 1 If l N, then f U l M k Γ 0 N/l,χ If N = l and χ is the trivial character, then f F l M k SL Z In the proof of congruence 11 on the generating function F m,k,r z, we need the following relation ηγz = ε a,b,c,d cz +d 1 ηz, 7 a b where γ = SL c d Z, ε a,b,c,d is a 4-th root of unity, and ηz is Dedekind s eta function as given by ηz = q q n 8 As a special case, we have n=1 η 1/z = z/i ηz 9 3 Generating functions of p r n modulo m k In this section, we derive the congruence of the generating function F m,k,r z defined by 111, namely, m k n+r F m,k,r z = p r q n 4 m k n r mod 4 6

7 Theorem 31 Let m 5 be a prime, and let k and r be positive integers Then there exists a modular form φ m,k,r z M λm,k,r SL Z, such that F m,k,r z η4z γ m,k,r φ m,k,r 4z mod m k, 31 where λ m,k,r = { m k m k 1 r γ m,k,r+r, if k is odd, m k m k 1 r γ m,k,r+r, if k is even, 3 γ m,k,r = 4β m,k,r r m k, 33 and β m,k,r is the unique integer in the range 0 β m,k,r < m k congruent to r/4 modulo m k The first step of the proof of Theorem 31 is to express F m,k,r z in terms of a modular form Consider the η-quotient r ηm k z mk f m,k,r z =, 34 ηz Γ 0 m k, kr The following lemma shows m whichisacuspformins m k 1r that F m,k,r z can be obtained from f m,k,r z by applying a U-operator and a V-operator Lemma 3 Let m 5 be a prime, and let k and r be positive integers Then we have F m,k,r z = f m,k,rz U m k V 4 35 η4z mk r Proof Since we find p r nq n = n=0 f m,k,r z = q mk 1 4 r n=0 = q mk 1 4 r n=1 n=1 1 1 q n r, 1 1 q n 1 q mkn mk r r n=1 p r nq n 1 q mkn mkr n=0 Applying the operator U m k, we obtain k 1+4βm,k,r f m,k,r z U m k = p r m k n+β m,k,r q n+rm 4m k 1 q n mkr, 7 n=1 n=1

8 where 0 β m,k,r m k 1 is determined by 4β m,k,r r mod m k So we deduce that n=0 k 1+4βm,k,r p r m k n+β m,k,r q n+rm 4m k Applying the operator V 4, we get = f m,k,rz U m k n=1 1 qn mk r p r m k n+β m,k,r q 4n+4β m,k,r r m k = f m,k,rz U m k V 4 36 η4z mk r n=0 Replacing 4n+ 4β m,k,r r m k by n in 36, or equivalently, n n 4 4β m,k,r r 4m k, one sees that the sum on the left hand side can be written in the form of F m,k,r z This completes the proof The second step of the proof of Theorem 31 is to derive a congruence relation for f m,k,r z U m k modulo m k Theorem 33 Let m 5 be a prime, and let k and r be positive integers Then there exists a modular form G m,k,r z M wm,k,r SL Z such that f m,k,r z U m k G m,k,r z mod m k, where w m,k,r = { m k m k 1 1 r, if k is odd, 3m k m k 1 1 r, if k is even Proof Let where ηz m g m,k,r z = ηmz ck m k 1 r 1, if k is odd, c k =, if k is even Since g m,k,r z is an η-quotient, using the modular transformation property due to Gordon, Hughes, and Newman[13,18,0], see also,[, Theorem 164], we deduce that g m,k,r z M ck m k m k 1 r 8 kr Γ 0 m, m,

9 Moreover, since 1 q mn 1 q n m mod m, we see that Since f m,k,r z S m k 1r we obtain that g m,k,r z 1 mod m k 37 f m,k,r z U m k 1 S m k 1r Γ 0 m k, kr, using Lemma 1 repeatedly, m kr Γ 0 m, m Thus, f m,k,r z U m k 1 g m,k,r z is a modular form on Γ 0 m of the trivial character and of weight w m,k,r = c km k m k 1 r Invoking Lemma 1, we find that + mk 1r G m,k,r z = f m,k,r z U m k 1 g m,k,r z F m 38 is a modular form in M wm,k,r SL Z where To complete the proof of Theorem 33, it remains to show that f m,k,r z U m k 1 g m,k,r z F m f m,k,r z U m k mod m k, 39 F m = U m +m w m,k,r 1 W m, and the operator W m is given by By congruence 37, we see that the left hand side of 39 equals f m,k,r z U m k +m w m,k,r 1 f m,k,r z U m k 1 g m,k,r z W m mod m k To prove 39, it suffices to show that m w m,k,r 1 f m,k,r z U m k 1 g m,k,r z W m 0 mod m k 310 We only consider the case when k is odd The case when k is even can be dealt with in the same manner In light of the transformation formula 9 of the eta function, we find that g m,k,r z W m = m mk m k 1 r 4 mz mk m k 1 r g m,k,r 1 mz = m mk m k 1 r 4 z mk m k 1 r mz/iηmz mk 1r m z/iηz 9

10 = m m+1mk 1 r 4 i m 1mk 1 r ηmz m ηz m k 1 r Therefore, 310 can be deduced from the following congruence m 3mk 1r 4 1 f m,k,r z U m k 1 W m 0 mod m k 311 By the property of U-operator as in 5, we have m 3mk 1r 4 1 f m,k,r z U m k 1 W m = m k+mk r r+4k 4 = m k+mk r r+4k 4 m k 1 1 μ=0 m k 1 1 μ=0 f m,k,r z m k 1r f m,k,r z m k 1r μ=0 1 μ 0 m k 1 μm 1 m k 0 Wm 31 Using the transformation formula 9 of the eta function, 31 can be written as m k 1 1 r m mk r k z mk 1r ηmμ 1 z mk η mμz 1 m k z m k 1 1 = m mk r k z r ηz m k r where α μ is a certain 4-th root unity μ=0 α μ η mμz r, m k z For μ = 0, we write μ = m s t where m t For μ = 0, we set s = k 1 and t = 0 Ineither case, thereexist integersbanddsuchthatbt+dm k s 1 = 1 It follows that mμ 1 m k 0 = t d m k s 1 b m s+1 b 0 m k s 1 Applying the corresponding slash operator to ηz, we obtain that mμz 1 η = ε m k μ m s+1 1 m s+1 z +b z η, z m k s 1 where ε μ is a 4-th root of unity Since the coefficients of the Fourier expansion of ηz at are integers and the coefficient of the term with the lowest degree is 1, the Fourier coefficients of each term in 313 are divisible by m mk s 1 r k in the ring Z[ζ 4 ] Clearly, 0 s k 1 Thus we have m k s 1 r k mk k r k mk k k k, 10

11 for m 5 and k 1 Hence the Fourier coefficients of each term in 313 are divisible by m k So we arrive at 311 This completes the proof We are now in a position to finish the proof of Theorem 31 Proofof Theorem31 ByTheorem33,thereexistsamodularformG m,k,r z M wm,k,r SL Z such that Let f m,k,r z U m k G m,k,r z mod m k 314 φ m,k,r z = G m,k,rz Δz mk r+γ m,k,r 4 where Δz = ηz 4 is Ramanujan s Δ-function In the proof of Lemma 3 we have shown that k 1+4βm,k,r f m,k,r z U m k = p r m k n+β m,k,r q n+rm 4m k 1 q n mkr, n=0 which implies that the order of the Fourier expansion of f m,k,r z U m k at is at least rm k 1+4β m,k,r = mk r+γ m,k,r 4m k 4 Thus φ m,k,r z is a modular form in M λm,k,r SL Z Combining 314 and Lemma 3, we conclude that F m,k,r z Δz m k r+γm,k,r 4 φ m,k,r z V 4 η4z mk r, n=1 as required = η4z γ m,k,r φ m,k,r 4z mod m k, 4 Congruences of p r n modulo m k In this section, we apply Theorem 31 on the congruence relation for the generating function F m,r,k z and Yang s method [31] to derive two classes of congruences of p r n modulo m k Let S γ,λ = {η4z γ φ4z: φz M λ SL Z} Yang [31] showed that when γ is anoddinteger such that 0 < γ < 4and λ is anonnegativeeveninteger, S γ,λ isaninvariantsubspaceofs λ+γ/ Γ 0 576,χ 1 under the action of the Hecke algebra More precisely, for all primes l =,3 and all f S γ,λ, we have f T l S γ,λ By the invariant property of S γ,λ, we obtain two classes of congruences of p r n modulo m k 11

12 Theorem 41 Let m 5 be a prime, k be a positive integer, r be an odd positive integer less than m k, and l be a prime different from,3 and m Then there exists an explicitly computable positive integer K such that p r m k l μk 1 n+r 4 0 mod m k 41 for all positive integers μ and all positive integers n relatively prime to l There is also a positive integer M such that m k l i n+r m k l M+i n+r p r p r mod m k for all nonnegative integers i and n ProofAccordingtocongruencerelation31,thegeneratingfunctionF m,k,r z is congruent to a modular form in S γm,k,r,λ m,k,r, where λ m,k,r and γ m,k,r are integers as given in 3 and 33 Let {f 1 z,,f d z} be a Z-basis of the space S γm,k,r,λ m,k,r Z[[q]] and f i z = n 0a i nq n, where i = 1,,d and q = e πiz To prove 41, it suffices to show that there exists a positive integer K such that m k l μk 1 n+r a i 0 mod m k 43 4 for all n coprime to l and i = 1,,d From the relation γ m,k,r m k = 4β m,k,r r, one sees that γ m,k,r and r have the same parity Since r < m k is odd, we have 0 < γ m,k,r < 4, and hence S γm,k,r,λ m,k,r is invariant under the Hecke operator T l So there exists a d d matrix A such that Let X = f 1 f d T l = A f 1 f d A I d l γ m,k,r+λ m,k,r I d 0 44 Using the property of the basis {f 1 z,,f d z} under the action of the U-operator as given by Yang [31, Corollary 34], we obtain f 1 f 1 g 1 f 1 U s l = A s +B s +C s V l, 45 f d f d 1 g d f d

13 where s is a positive integer, g i = f i l, and As,B s and C s are d d matrices given by As A s 1 = Id 0 X s, 46 B s = l λ m,k,r+γ m,k,r 3/ 1 γ m,k,r 1/ 1 A s 1, l C s = l γ m,k,r+λ m,k,r A s 1 Since gcdm,l = 1, the matrix X mod m k is invertible in the ring M consisting of d d matrices over Z m k By the finiteness of M, we see that there exist integers a > b such that X a and X b are linear dependent over Z m k, ie, there exists a constant c Z m k such that X a cx b mod m k Thus X K ci d mod m k, where K = a b In view of the relation AμK 1 A μk c μ I d 0 X 1 mod m k, we find that A μk 1 0 mod m k Hence, from 45 it follows that f 1 g 1 f 1 U μk 1 l B μk 1 +C μk 1 V l mod m k f d g d f d Applying the U-operator U l to both sides and observing that Ul g i U l = f i = 0, l the relation 45 leads to the following congruence f 1 f 1 U μk 1 l U l C μk 1 V l mod m k, f d f d which implies 43 We now turn to the proof of congruence 4 By the finiteness of M, we see that there exists a positive integer M such that X M I d mod m k Thus matrix equation 46 reduces to the following congruence AM A M 1 Id 0 mod m k It follows that A M I d mod m k and B M C M 0 mod m k Thus, relation 45 implies f 1 f 1 U M l mod m k f d f d 13

14 So the coefficient of q n is congruent to the coefficient of q lmn in f i z modulo m k forall iandn Since F m,k,r z is a linear combination off i z with integer coefficients, we obtain congruence 4 This completes the proof 5 Examples In this section, we present some consequences of Theorem 31 and Theorem 41 We first give some examples for the congruences of the generating function F m,k,r z of p r n Example 51 By Theorem 31, we find F m,k,r z η4z γ m,k,r φ m,k,r 4z mod m k, where γ m,k,r is an integer, φ m,k,r z is a polynomialof Δz and the Eisenstein series E 4 z and E 6 z Table 51 gives the list of explicit expressions of ηz γ m,1,r φ m,1,r z for m 19 and r 7 r m ηz γ m,1,r φ m,1,r z ηz ηz E 4 z 13 8ηz 17 5ηz 14 E 4 z 19 ηz 10 14E 4 z 3 +1Δz 3 5 4ηz 9 7 3ηz 3 E 6 z ηz 9 4E 4 z 3 +6Δz ηz 15 E 6 z 3 +3E 6 zδz 4 5 4ηz 4 E 4 z ηz 4 3E 4 z 4 +8E 4 zδz 13 ηz 0 7E 4 z 3 +4Δz 17 ηz 4 6E 4 z 7 +11E 4 z 4 Δz+4E 4 zδz 19 ηz 0 16E 4 z 6 +18E 4 z 3 Δz+Δz 5 5 ηz 1 E 4 z 7 ηz 13 E 6 z ηz 7 8E 4 z 6 +11E 4 z 3 Δz+5Δz 17 ηz 11 16E 4 z 8 +16E 4 z 5 Δz+4E 4 z Δz 19 ηz5e 6 z 7 +15E 6 z 5 Δz+16E 6 z 3 Δz 14

15 ηz 6 6E 4 z 3 +6Δz 11 ηz 6 10E 4 z 6 +E 4 z 3 Δz 13 ηz 18 7E 4 z 6 +8E 4 z 3 Δz+6Δz 17 ηz 18 3E 4 z 9 +3E 4 z 6 Δz+5E 4 z 3 Δz 19 ηz 6 6E 4 z 1 +E 4 z 9 Δz+14Δz ηz 1 E 6 z ηz 5 10E 4 z 9 +6E 4 z 6 Δz+9E 4 z 3 Δz +11Δz 3 17 ηz7e 4 z 13 +E 4 z 10 Δz+E 4 z 7 Δz +3E 4 z 4 Δz Table 51: Explicit congruences derived from Theorem 31 Example 5 Let 0 β < m k be an integer with β r/4 mod m k If φ m,k,r z 0 mod m k, using Theorem31 weobtain the followingramanujantype congruences of multipartition functions p r m k n+β 0 mod m k 51 The values of m and β for r 9 and k = 1, are given in Table 5 r m,β m,β 1 5,4,7,5,11,6 5,4,49,47,11,116 5,3 5,3 3 11,7,17,15 11, ,6 49, ,8,3,5 11,96 6 5,4 5,19 7 5,3,11,9,19,9 5,18,11,86 8 7,5,11,4 11, ,11,19,17,3,9 Table 5: Ramanujan-type congruences of multipartitions It can be seen that Table 5 contains the Ramanujan congruences 11 of pn modulo 5,7 and 11, as well as Gandhi s congruences 14 for p n and 15 for p 8 n The following examples demonstrate how to derive certain congruences of p r n with the aid of Theorem 41 15

16 Example 53 For the values of l and K l as given in Table 53, we have 7 l μk l 1 n+3 p 3 0 mod for all positive integers μ and all positive integers n not divisible by l l a l K l Table 53: Eigenvalues a l of F 7,1,3 z acted by T l and the corresponding K l Proof By Theorem 31 we find F 7,1,3 z 3η4z 3 E 6 4z mod 7 Since η4z 3 E 6 4z belongs to the 1-dimensional space S 3,6, for any prime l =,3,7, there exists an integer a l such that F 7,1,3 z T l a l F 7,1,3 z mod 7 Inspecting the proof of Theorem 41, we obtain the corresponding orders K l for which congruence 5 holds Example 54 We have p n mod 5 for all integers n coprime to 13 and 5 13 i n i n+3 p 3 p mod 5 for all nonnegative integers n and i Proof By Theorem 31, F 5,,3 z is congruent to a modular form in the space S 1,48 of dimension 5 Setting f i = η4z 1 E 4 4z 35 i Δ4z i 1, for 1 i 5 Clearly, f 1,f,,f 5 form a Z-basis of S 1,48 Z[[q]] Let A be the matrix of T l with respect to this basis By computing the first five 16

17 Fourier coefficients of f i and f i T 13 and equating the Fourier coefficients of both sides of 44, we find A mod 5, with the corresponding orders K = M = 100 Setting μ = 1 in Theorem 41, we complete the proof Below are two more examples for p 3 n and p 5 n modulo 7 The proofs are analogous to the proof of the above example, and hence are omitted Example 55 We have p n mod 7 for all positive integers n coprime to 7 and 7 11 i n i n+3 p 3 p mod 7 for all nonnegative integers n and i Example 56 We have p n mod 7 for all positive integers n coprime to 17 and 7 17 i n i n+5 p 5 p mod 7 for all nonnegative integers n and i Acknowledgments This work was supported by the 973 Project, the PC- SIRT Project of the Ministry of Education, and the National Science Foundation of China 17

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