Some generalizations of a supercongruence of van Hamme

Some generalizations of a supercongruence of van Hamme Victor J. W. Guo School of Mathematical Sciences, Huaiyin Normal University, Huai an, Jiangsu 3300, People s Republic of China jwguo@hytc.edu.cn Abstract. van Hamme conjectured that, for any odd prime p, 4 + 1 64) ) 3 p 1) mod p 3 ). This result was proved by Mortenson using a 6 5 transformation, and was reproved by Zudilin via the Wilf Zeilberger method. In this paper, we propose a conjectural generalization of van Hamme s supercongruence and prove some special cases, such as 4 + 1) 3 64) ) 3 3p 1) mod p 3 ), where p is a prime with p 3 mod 8). We also state some related conjectures. Keywords: supercongruence; gamma function; Whipple s 7 6 transformation; Wilf Zeilberger method; WZ-pair. 010 Mathematics Subject Classifications: 33C0, 33B15, 11A07, 11B65, 65B10 1. Introduction In 1997, van Hamme [1] conjectured that Ramanujan s formula for 1/π []: has a nice p-adic analogue: ) 3 4 + 1 64) π, 1.1) 4 + 1 64) ) 3 p 1) mod p 3 ), 1.) where p is an odd prime. In 008, the congruence 1.) was first proved by Mortenson [3] using an idea of McCarthy and Osburn [4] and a 6 5 transformation. In 009, Zudilin [5] 1

reproved 1.) via the Wilf Zeilberger method. In 013, Z.-W. Sun [6] gave a related congruence by the Wilf Zeilberger method again. Recently, motivated by Zudilin s wor, the author [7] has given a q-analogue of 1.) by the q-wilf Zeilberger method, along with the following generalization of 1.): 4 + 1 64) ) 3 p r 1) )r mod p r+ ). 1.3) This paper was motivated by the following further generalization of 1.3). Conjecture 1.1. or any odd prime p and positive odd integer m, there exists an integer a m such that, for any positive integer r, there hold 4 + 1) m 64) 4 + 1) m 64) ) 3 a m p r 1) )r mod p r+ ), 1.4) ) 3 a m p r 1) )r mod p r+ ). 1.5) In particular, we have a 1 1, a 3 3, a 5 41, a 7 1595, a 9 14689 and a 11 1653107. or m 3, we have no formula involving 4 +1) m similar to 1.1). In fact, for m 3, we have ) 4 + 1) m 3. 64) The objective of this paper is to prove some special cases of Conjecture 1.1. Our first result can be stated as follows. Theorem 1.. Let p be an odd prime. Then 4 + 1) 3 64) urthermore, for p 3 mod 8), we have ) 3 3p 1) mod p ). 1.6) 4 + 1) 3 64) ) 3 3p 1) mod p 3 ). 1.7) Our second result is on a supercongruence involving 4 + 1) 5.

Theorem 1.3. Let p be an odd prime such that 19 + 176 + 41 0 mod p) for 0, 1,...,. Then 4 + 1) 5 64) ) 3 41p 1) mod p 3 ). Our third result is on a supercongruence involving 4 + 1) 7. Theorem 1.4. Let p be an odd prime such that 7680 3 + 14016 + 880 + 1595 0 mod p) for 0, 1,...,. Then 4 + 1) 7 64) ) 3 1595p 1) mod p 3 ). 1.8) Note that van Hamme [1] also made some other similar conjectures on supercongruences, such as 6 + 1 56 ) 3 p 1) mod p 3 ), which was proved by Long [8]. However, we did not find any similar generalizations of these supercongruences. The paper is organized as follows. In Section, we give a proof of 1.6) by a hypergeometric identity. In Section 3, we give proofs of Theorems 1. 1.4 by the Wilf Zeilberger method. We give some related conjectures on supercongruences in Section 4.. Proof of 1.6) by a hypergeometric identity We first give some bacground information. The gamma function Γz), for any complex number z with the real part positive, is defined as Γz) 0 x z 1 e x dx, and can be uniquely analytically extended to a meromorphic function defined for all complex numbers z, except for integers less than or equal to zero. It is well-nown that the gamma function satisfies the property Γz + 1) zγz). We shall also use a hypergeometric identity, which is a specialization of Whipple s 7 6 transformation see [9, p. 8]): [ ] a, 1 + 1a, b, c, d, e 6 5 ; 1 a, 1 + a b, 1 + a c, 1 + a d, 1 + a e 1 Γ1 + a d)γ1 + a e) Γ1 + a)γ1 + a d e) 3 [ ] 1 + a b c, d, e 1 + a b, 1 + a c ; 1,.1) 3

where [ ] a0, a 1,..., a r r+1 r ; z b 1,..., b r a 0 ) a 1 ) a r )!b 1 ) b r ) z, and a) n aa + 1) a + n 1) for n 0. ollowing the argument of Mortenson [3], we use an idea of McCarthy and Osburn [4] to mae the following choice of variables in.1). We let a 1, b 5, c 5, d 1 p 4 4 and e 1+p to produce 6 5 [ 1, 5 4, 5, 5, 1 p, 1+p 4 4 1, 1, 1, 1 + p, 1 p 4 4 4 ; 1 It is easy to see that, for 0, 1,..., 1+p ) ) 1 p 1 p ) ) 1 + p i1 ], Γ1 + p )Γ1 p ) Γ 3 )Γ 1 ) 3 i 1) p 4i p By using the property Γx + 1) xγx), we have and so as desired. 4 + 1) 3 64) Γ1 + p )Γ1 p ) Γ 3 )Γ 1 ) p 1), ) [ 3 1, p 1) ) 3 p 1) ) 1 41 p )) 3p 1) ) mod p ), 1 ) mod p ).! 1 p, 1+p 1, 1 4 4 3. Proofs by the Wilf Zeilberger method [ 1, 1 p, 1+p ; 1 ] 1, 1 4 4 Proof of Theorem 1.. or all non-negative integers n and, define the functions n, ) 1) n+ 34n + 1)3 1 ) n 1) n+ 8 + 3)1) n1) n 1, ) Gn, ) 1) n+ 618n 80n 64n + 40n 40 + 7) 1 ) n 1) n+ 1 8 5)8 + 3)1) n 11) n 1, ) where 1/1) m 0 for m 1,,.... The functions n, ) and Gn, ) form a WZ-pair. Namely, they satisfy the relation n, 1) n, ) Gn + 1, ) Gn, ). 3.1) ; 1, ]. 4

Indeed, we have n, 1) Gn, ) 1 ) 8 + 3)4n + 1) 3 n + 1)18n 80n 64n + 40n 40 + 7)n, n, ) Gn, ) n + 1 )8 5)4n + 1)3 18n 80n 64n + 40n 40 + 7)n, Gn + 1, ) Gn, ) It is routine to verify that n + 1 ) n + 1 )18n + 19n + 4 80n 10n 13) n + 1)18n 80n 64n + 40n 40 + 7)n. n, 1) Gn, ) n, ) Gn, ) i.e., the identity 3.1) holds. Summing 3.1) over n from 0 to, we obtain n, 1) urthermore, for 1,,...,, there holds ) p + 1 G, Gn + 1, ) Gn, ) 1, ) ) p + 1 p + 1 n, ) G, G0, ) G,. 3.) 1) p+1)/+ 63p + 3p 40 0p 0p + 7)p 1 ) )/ 1 ) p+1)/+ 1 8 5)8 + 3)1) )/ 1) p+1)/ 1 ) 0 mod p ), 3.3) since 1/) p+1)/+ 1 is divisible by p, while the p-adic order of the denominator is at most 1. Combining 3.) and 3.3), we see that n, 0) n, 1) n, ) n, p 1 ) mod p ). 3.4) On the other hand, we have n, p 1 ) p 1, p 1 ) 3p 1)3 1) 4p 1)1) )/ 3p 1) 4p 1) ) p p 1 p 1 ) p 1 ). 5

Note that p 1) 4p 1 1 mod p ). It remains to use Babbage s congruence see [10]) ) p 1 1 mod p ) 3.5) p 1 and Morley s congruence see [11]) ) p 1 to conclude that 1) 4 n, p 1 ) 3p 1) mod p 3 ). mod p ) 3.6) The congruence 1.6) then follows from 3.4). We now suppose that p 3 mod 8). Then for 1,,..., we have 8 5)8 + 3) 0 mod p), and the congruences 3.3) and 3.4) hold modulo p 3. Similarly as before, we immediately obtain 1.7). Proof of Theorem 1.3. The rational functions n, ) 1) n+ 414n + 1) 5 1 ) n 1) n+ 19 + 176 + 41)1) n1) n 1, ) Gn, ) 1) n+ 4915n 4 5348n 4 + 1459n 4 4915n 3 + 5348n 3 1459n 3 664n + 3768n 10144n + 19456n 3040n + 6896n + 818 1035 + 331) 8 1 ) n 1) n+ 1 19 08 + 57)19 + 176 + 41)1) n 11) n 1. ) satisfy 3.1). Suppose that p is an odd prime such that 19 + 176 + 41 0 mod p) for 0, 1,...,. Then for 1,..., n, 1) 19 08 + 57)19 + 176 + 41) 0 mod p) 3.7). Similarly as before, summing 3.1) over n from 0 to, we obtain ) ) p + 1 p + 1 n, ) G, G0, ) G,. 3.8) 6

urthermore, for 1,,...,, in view of 3.7), we have ) p + 1 G, 0 mod p 3 ), 3.9) Combining 3.8) and 3.9), we are led to Moreover, n, 0) n, 1) n, p 1 ) p 1, p 1 ) n, p 1 ) 41p 1) 5 1 ) 48p 8p + 1)1) )/ 41p 1)4 p 1 48p 8p + 1) ) p p 1 It is easy to see that p 1) 4 48p 8p + 1) 1 mod p ). Substituting this into 3.11) and using 3.5) and 3.6), we conclude that The proof then follows from 3.10). n, p 1 ) 41p 1) mod p 3 ). mod p 3 ). 3.10) ) p 1 ). 3.11) Proof of Theorem 1.4. following This time the selection of a WZ-pair that satisfies 3.1) is the n, ) 1) n+ 15954n + 1) 7 1 ) n 1) n+ 7680 3 + 14016 + 880 + 1595)1) n1) n 1, ) Gn, ) 1) n+ 3145780 3 n 6 4718590 3 n 5 3696304 n 6 7918336 3 n 4 + 55443456 n 5 + 13467648n 6 + 47579136 3 n 3 + 46940160 n 4 00147n 5 149504n 6 + 1590067 3 n 70041600 n 3 633938n 4 + 14456n 5 1438948 3 n 8658688 n + 34756608n 3 + 4939008n 4 538984 3 + 366080 n + 1757600n 583448n 3 + 9594816 13183808n 3475504n 1 5755064 + 489144n + 116755) ) n 1) n+ 1 1) n 11) n 1 ) 3190 7680 3 904 + 388 349)7680 3 + 14016 + 880 + 1595). 7

With the assumption that 7680 3 + 14016 + 880 + 1595 0 mod p) for 0, 1,...,, we have 7680 3 904 + 388 349)7680 3 + 14016 + 880 + 1595) 0 mod p) for 1,...,. It follows that G p+1, ) 0 mod p 3 ) for 1,...,. By applying 3.1), we conclude that n, 0) n, p 1 ) mod p 3 ). 3.1) urthermore, n, p 1 ) inally, noticing that 1595p 1) 6 960p 3 + 64p + 1p 1) ) p p 1 p 1 p 1) 6 960p 3 + 64p + 1p 1 1 mod p ), ) p 1 ). 3.13) and using 3.5), 3.6), 3.1), and 3.13), we finish the proof of 1.8). 4. Concluding remars and open problems Numerical calculation suggests the following conjecture, which has been verified for the first 10 5 primes. Conjecture 4.1. or any odd prime p and 0, 1,...,, there hold 19 + 176 + 41 0 mod p ), 4.1) 7680 3 + 14016 + 880 + 1595 0 mod p ). 4.) By the proof of Theorem 1.3, if the congruence 4.1) is true, then we have 4 + 1) 5 64) ) 3 41p 1) mod p ), since 19 08 + 57 19 1) + 176 1) + 41 and it is easily seen that gcd19 08 + 57, 19 + 176 + 41) 1, 73. 8

Similarly, if the congruence 4.) is true, then 4 + 1) 7 64) ) 3 1595p 1) mod p ). Note that, by 1.3), the congruence 1.4) is true for m 1 by taing a 1 1, while the congruence 1.5) is still open for m 1, except for r 1. If Conjecture 1.1 is true for m 1, 3, then 4 + 1)4 + + 1) 64) ) 3 0 mod p r+ ), since 44 + 1)4 + + 1) 4 + 1) 3 + 34 + 1). It seems that we have the following much more stronger conjecture. Conjecture 4.. Let p be an odd prime and r a positive integer. Then 4 + 1)4 + + 1) 64) 4 + 1)4 + + 1) 64) Long [8, Theorem 1.1] proved that ) 3 p 3r mod p 3r+1 ), ) 3 p 3r mod p 3r+1 ). 4 + 1 56 ) 4 p mod p 4 ) 4.3) for any prime p > 3. It should be mentioned that Long [8, Theorem 1.1] in fact gave the following generalization of 4.3): 4 + 1 56 ) 4 p r mod p r+3 ). 4.4) However, her proof is valid only for r 1. The reason is that she used McCarthy and Osburn s argument incorrectly. Specifically, from A B mod p 4r ) and C D mod p 4r ) we cannot deduce that A/C B/D mod p 4r ). To see this, tae A C D p and B p + p 4r. Thus, the supercongruence 4.4) is still a conjecture up to now. In what follows, we shall give a further generalization of 4.4) similar to 1.4) and 1.5). 9

Conjecture 4.3. or any prime p > 3, positive odd integer m and positive integer r, there hold 4 + 1) m 56 4 + 1) m 56 ) 4 b m 1) m 1 p r mod p r+3 ), ) 4 b m 1) m 1 p r mod p r+3 ), where b m is the coefficient of x m 1 in the expansion ) exp 1) n x n E n, n n1 and E n is the n-th Euler number. or example, b 1 1, b 3 1, b 5 3, b 7 3, b 9 371 and b 11 10515 see the sequence A55881 in the OEIS). We also give another difficult conjecture as follows. Conjecture 4.4. Let p be an odd prime and r a positive integer. Then 4 + 1)8 + 4 + 1) 56 4 + 1)8 + 4 + 1)) 56 ) 4 p 4r mod p 4r+1 ), ) 4 p 4r mod p 4r+1 ). In Ramanujan s second letter to Hardy on ebruary 7, 1913, he mentioned the following identity ) 5 1) 4 + 1) 4 5 Γ 3, 4.5) )4 4 A p-adic analogue of 4.5) was also conjectured by van Hamme [1] as follows: ) 5 4 + 1 p Γ p 3 mod p 3 ), if p 1 mod 4), 104) 4 )4 0 mod p 3 ), if p 3 mod 4), 4.6) where p is an odd prime and Γ p ) is the p-adic gamma function. The congruence 4.6) was later proved by McCarthy and Osburn [4]. We have the following conjecture, which is related to 4.6). 10

Conjecture 4.5. Let p be a prime of the form 4 + 1 and r a positive integer. Then 4 + 1)8 + 4 + 1) 104) 4 + 1)8 + 4 + 1) 104) ) 5 0 mod p 3r ), ) 5 0 mod p 3r ). or general m, it seems that we cannot unify Conjecture 1.1 and [1, B.3)] of Swisher, or Conjecture 4.3 and [1, C.3)]. or m 1, we have the following accompanying conjecture for [1, B.3) and C.3)]. Conjecture 4.6. Let p be a prime and let r be a positive integer. Then 4 + 1 64) 4 + 1 56 ) 3 p r 1 1 p 1) ) 4 p r 1 1 p 4 + 1 56 4 + 1 64) ) 3 mod p 3r ), ) 4 mod p 4r ), p > 3. Acnowledgments. The author would lie to than the anonymous referee for many helpful comments and suggestions. This wor was partially supported by the National Natural Science oundation of China grant 11371144), the Natural Science oundation of Jiangsu Province grant BK0161304), and the Qing Lan Project of Education Committee of Jiangsu Province. References [1] van Hamme L. Some conjectures concerning partial sums of generalized hypergeometric series. in: p-adic unctional Analysis Nijmegen, 1996), Lecture Notes in Pure and Appl. Math. 19. Deer, New Yor; 1997;3 36. [] Ramanujan S. Modular equations and approximations to π. Quart. J. Math. Oxford Ser. ) 1914;45:350 37; Reprinted in Collected papers of Srinivasa Ramanujan, G.H. Hardy, P.V. Sechu Aiyar, and B.M. Wilson eds.), Cambridge University Press, Cambridge, 197; Chelsea Publ., New Yor, 196;3 39. [3] Mortenson E. A p-adic supercongruence conjecture of van Hamme. Proc. Amer. Math. Soc. 008;136:431 438. [4] McCarthy D, Osburn R. A p-adic analogue of a formula of Ramanujan. Arch. Math. 008;91:49 504. [5] Zudilin W. Ramanujan-type supercongruences. J. Number Theory. 009;19:1848 1857. [6] Sun Z-W. Products and sums divisible by central binomial coefficients. Electron. J. Combin. 013;01):#P9. 11

[7] Guo VJW. A q-analogue of a Ramanujan-type supercongruence of van Hamme and Mortenson. J. Math. Anal. Appl., to appear. [8] Long L. Hypergeometric evaluation identities and supercongruences. Pacific J. Math. 011;49:405 418. [9] Bailey WN. Generalized Hypergeometric Series. Cambridge Tracts in Mathematics and Mathematical Physics 3, Cambridge University Press, London; 1935. [10] Babbage C. Demonstration of a theorem relating to prime numbers. Edinburgh Philos. J. 1819;1:46 49. [11] Morley. Note on the congruence 4n 1) n n)!/n!), where n + 1 is a prime. Ann. of Math. 1895;9:168 170. [1] Swisher H. On the supercongruence conjectures of van Hamme. Res. Math. Sci. 015;:18. 1