CONGRUENCES SATISFIED BY APÉRY-LIKE NUMBERS

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1 International Journal of Number Theory Vol 6, No 1 ( c World Scientific Publishing Comany DOI: /S CONGRUENCES SATISFIED BY APÉRY-LIKE NUMBERS HENG HUAT CHAN, SHAUN COOPER FRANCESCO SICA Deartment of Mathematics, National University of Singaore Block S17, 10, Lower Kent Ridge Road, Singaore matchh@nusedusg Institute of Information Mathematical Sciences Massey University, Private Bag North Shore Mail Centre, Auckl, New Zeal scooer@masseyacnz Mathematics Comuter Science, Mount Allison University 67 York Street, Sackville, NB, E4L 1E6, Canada fsica@mtaca Received 12 May 2008 Acceted 16 June 2008 In this article, we investigate congruences satisfied by Aéry-like numbers Keywords: Aéry numbers; congruence; modular form Mathematics Subect Classification 2010: 11B83, 11A07 1 Introduction: Aéry Numbers In his roof of the irrationality of ζ(3, R Aéry introduced the numbers n ( 2 ( 2 n n + α n =, n N ThesenumbersarenowknownastheAéry numbers Since the aearance of Aéry s work, roerties of α n were gradually discovered One of these is the observation that for rimes 5, α α 1 (mod 3 (11 The congruence (11 was conectured by Chowla et al [6] roved by Gessel [7], who established the stronger result α n α n (mod 3 (12 In this article, we investigate other sequences of integers {f n } n=1 relations similar to (12 that satisfy 89

2 90 H H Chan, S Cooer & F Sica Let η(τ =q 1/24 n=1 (1 q n, where q =ex(2πiτ Im(τ > 0 It can be shown [10] thatif ( 12 η(6τη(τ t 1 (τ = F 1 (τ = η7 (2τη 7 (3τ η(2τη(3τ η 5 (τη 5 (6τ, F 1 (τ = α n t n 1 (τ (13 for suitably small t 1 (τ The identification of α n as the coefficients of certain ower series serves as a starting oint for us in our search of other sequences {f n } n=1 satisfying congruences similar to (12 2 The Domb Numbers Consider the functions t 2 (τ = ( η(2τη(6τ η(τη(3τ 6 F 2 (τ = (η(τη(3τ4 (η(2τη(6τ 2 It can be shown [2, (414] that when t 2 (τ is sufficiently small, we have F 2 (τ = ( 1 n β n t n 2 (τ (21 where β n = n ( n 2 ( 2 ( 2(n n The sequence {β n } n=1 turns out to satisfy the congruence Theorem 21 For rimes 5, β n β n (mod 3 Proof The method of roof given here is due to Gessel [7] For a rime 5, we find that n ( 2 ( ( n 2 2(n β n = n where S 1 = = S 1 + S 2, (22 n ( n 2 ( 2 ( 2(n (n

3 Congruences Satisfied by Aéry-Like Numbers 91 Now, since [8] S 2 = k=1 m=0 S 1 ( a b ( n k + m n ( n ( a b k=1 2 ( 2(k + m k + m 2 ( 2 ( 2(n n ( 2(n k m n k m (mod 3 (mod 3 for rimes 5 (23 Therefore, S 1 β n (mod 3 (24 For 0 <k<,wehave[7] ( n ( 1 k n ( n 1 k + m k m (mod 2 Hence, S 2 2 n 2 1 ( 2 n 1 k 2 m m=0 In order to rove that it suffices to show that 1 ( n 1 k 2 m m=0 k=1 ( 2k +2m k + m 2 ( 2k +2m k + m By Lucas congruence [9], ( a + b c + d Hence, we deduce that 1 ( 2 ( n 1 2k +2m k 2 k=1 m=0 m k + m n = k=1 k=1 1 k 2 1 k 2 m=1 n m=1 ( n 1 m 1 ( n 1 m 1 ( 2(n k m n k m S 2 0 (mod 3, ( a c ( 2(n k m n k m ( b d ( 2(n k m n k m 2 ( 2k +2(m 1 k + (m 1 2 ( 2k k ( 2(m 1 (m 1 (mod 3 (25 0 (mod (26 (mod (27 ( 2(n k (m 1 n k (m 1 ( 2(n m n m ( 2( k k (mod

4 92 H H Chan, S Cooer & F Sica But for 1 k 1, Hence, we deduce (26 ( 2k k ( 2k k or ( 2( k k ( 2( k k 0 (mod, A simle corollary of Theorem 21 is that for all rime numbers >3 β β 1 4 (mod 3 3 Almkvist Zudilin Sequence The study of the sequence {β n } n=1 is insired by the fact that α n aears as the coefficients of the ower series given by (13 As we have seen above, β n are coefficients of the ower series given by (21 There is a third sequence that behaves similarly to both α n β n To motivate our discovery of this third sequence, we observe that F 1 F 2 are modular forms associated with Γ 0 (6 +6 Γ 0 (6 +3 resectively Naturally, one would exect to have a third sequence arising from Γ 0 (6 +2 Indeed, in [5] itwasshownthatif ( 4 η(3τη(6τ t 3 (τ = η(τη(2τ F 3 (τ = (η(τη(2τ3 η(3τη(6τ, t 3 (τ is sufficiently small, F 3 (τ = ( 1 n γ n t n 3 (τ, where γ n are the Almkvist Zudilin numbers [1], given by γ n = n/3 ( 1 3n 3 (3! (! 3 The numbers γ n aear to satisfy the congruence Conecture 31 for all rimes >3 γ n γ n (mod 3 ( ( n n + (31 3 We have been unable to give a roof of Conecture 31 as Gessel s method does not seem to work in this case

5 Congruences Satisfied by Aéry-Like Numbers 93 4 Yang Zudilin Sequence For ositive integers k n, let y k,n = n ( k n Around 2003, Zudilin realized that y 4,n is associated with a certain modular form modular function as in the case for the Aéry numbers, Domb numbers the Almkvist Zudilin numbers This form function were eventually obtained by Yang [11] (see[4] for the exlicit forms of the form function In this section, we will deduce that for rimes 7, y 4, y 4,1 2 (mod 5 by showing the following more general result: Theorem 41 Suose k is even, >3 is a rime number for which k Then y k, 2 (mod k+1 Proof Observe that ( for 1 1 Hence it suffices to show that ( k ( 1! (41!(! =1 Now ( 1!!(! = 1 i 1 i ( 1 (mod i=1 Thus, since k is even, ( k ( 1!!(! =1 But k =1 =1 1 k k (mod (42 =1 { 0 (mod, if 1 k, 1 (mod, if 1 k (43 By hyothesis 1 k, therefore (41 follows from (42 (43 This comletes the roof Theorem 41 does not have a generalization modulo k+1 similar to Theorem 21 However, we have the following result: Theorem 42 Let >3 be rime let k>1 be an integer Then y k,n y k,n (mod 3

6 94 H H Chan, S Cooer & F Sica Proof When k =2wehave so y 2,n = y 2,n = n ( 2n n ( 2 n = ( 2n, n ( 2n y 2,n (mod 3 n by (23 This establishes the result for k = 2 For the remainder of the roof, suose k 3write n ( k n y n = = T 1 + T 2, where Using (23, we deduce Next, we rewrite T 2 as T 2 = =1 m=0 T 2 = T 1 = n =1 m=0 ( k n, ( k n + m T 1 y n (mod 3 ( k n = + m =1 m=0 ( k + (n 1 (44 + m By (27, we find that ( ( ( + (n 1 n 1 0 (mod + m m This imlies that for k 31 1, ( k + (n 1 0 (mod 3 (45 + m Substituting (45 into(44, we conclude that T 2 0 (mod 3 this comletes the roof of Theorem 42 5 Other Sequences We hoe that we have illustrated that sequences arising from the study of modular forms serve as a good source of numbers satisfying interesting congruences modulo

7 Congruences Satisfied by Aéry-Like Numbers 95 certain ower of rimes We end this article with a series of conectures associated with various modular forms The letter always denotes a rime number Conecture 51 If z 2 = Conecture 52 If z 3 = Conecture 53 If z 2 = q m2 +n 2 x 2 = η12 (2τ z 6 2 f 2,n x n 2, f 2,n f 2,n (mod 2 when 1 (mod 4 q m2 +mn+n 2 x 3 = η6 (τη 6 (3τ z 6 3 z 3 = f 3,n x n 3, f 3,n f 3,n (mod 2 when z 5 = η5 (τ η(5τ z 5 = f 5,n x n 5, x 5 = η6 (5τ η 6 (τ ( =1 3 f 5,n f 5,n (mod 3 for all rimes, including =2 Conecture 54 If z 7 = q m2 +mn+2n 2 x 7 = η3 (τη 3 (7τ z 3 7 z 7 = f 7,n x n 7, ( f 7,n f 7,n (mod 2 when =1 7

8 96 H H Chan, S Cooer & F Sica Conecture 55 If z 11 = Conecture 56 If z 23 = q m2 +mn+3n 2 x 11 = η2 (τη 2 (11τ z 2 11 z 11 = f 11,n x n 11, ( f 11,n f 11,n (mod 2 when =1 11 q m2 +mn+6n 2 x 23 = η(τη(23τ z 23 z 23 = f 23,n x n 23, f 23,n f 23,n (mod ( when =1 23 Conecture 57 If Z 23 = q 2m2 +mn+3n 2 X 23 = η(τη(23τ Z 23 Z 23 = F 23,n X23, n F 23,n F 23,n (mod ( when =1 23 Remarks One can verify that ( 1 2 ( 1 ( 1 f 2,n =64 4 n n (n! 2 f 3,n = n n 3 n (n! 2, where (a k = a(a+1(a+2 (a+k 1 There are no known closed forms for f r,n for r =5, 7, but they satisfy certain recurrence relations The functions z r x r,forr = 3, 7, 11 23, were studied in [3]

9 Congruences Satisfied by Aéry-Like Numbers 97 Acknowledgments We thank W Zudilin for correcting the roof of Theorem 42 given in an earlier version of this article The first author is suorted by National University of Singaore Academic Research Fund R References [1] G Almkvist W Zudilin, Differential equations, mirror mas zeta values, in Mirror Symmetry V, AMS/IP Stud Adv Math, Vol 38 (Amer Math Soc, Providence, RI, 2006, [2] H H Chan, S H Chan Z-G Liu, Domb s numbers Ramanuan Sato tye series for 1/π, Adv Math 186 ( [3] H H Chan S Cooer, Powers of theta functions, Pacific J Math 235 ( [4] H H Chan, Y Tanigawa, Y F Yang W Zudilin, New analogues of Clausen s identities arising from the theory of modular forms, rerint [5] H H Chan H Verrill, The Aéry numbers, the Almkvist Zudilin numbers new series for 1/π, Math Res Lett 16(3 ( [6] S Chowla, J Cowles M Cowles, Congruence roerties of Aéry numbers, J Number Theory 12 ( [7] I Gessel, Some congruences for Aéry numbers, J Number Theory 14 ( [8] G S Kazzidis, Congruences on the binomial coefficients, Bull Soc Math Grèce (NS 9 ( [9] E Lucas, Sur les congruences des nombres eulériens et les coefficients différentiels des fonctions trigonométriques, suivant un module remier, Bull Soc Math France 6 ( [10] C Peters J Stienstra, A encil of K3-surfaces related to Aéry recurrence for ζ(3 Fermi surfaces for otential zero, in Arithmetic of Comlex Manifolds (Erlangen 1988, Lecture Notes in Mathematics, Vol 1399 (Sringer, 1989, [11] Y F Yang, Private communication to H H Chan (16 November 2005