Ramanujan-type supercongruences. (English) Zbl 1231.11147

The starting point of this paper comes from the observation that several of Ramanujan’s and Ramanujan–like formulas for \(1/\pi\) admit natural conjectural \(p\)-adic analogues. The thesis set forward in this paper is that perhaps all known Ramanujan’s formulas for \(1/\pi\) and even the ones for \(1/\pi^2\) admit conjectural \(p\)-adic congruences. The author presents several such conjectural congruences and proves three of them. The proofs use the WZ–method of creative telescoping together with some classical \(p\)-adic congruences such as Wolstenholme’s theorem. The paper concludes with an experimental section presenting several conjectural cogruences modulo \(p^3,~p^5\) or higher powers of \(p\), of which the author can prove only a few usually modulo a smaller power of \(p\) than experiments seem to indicate. Due to the abundance of conjectural supercongruences it is likely that this paper will spur a lot of interest in the area of \(p\)-adic Ramanujan supercongruences.


11Y55 Calculation of integer sequences
33C20 Generalized hypergeometric series, \({}_pF_q\)
11B65 Binomial coefficients; factorials; \(q\)-identities
11F33 Congruences for modular and \(p\)-adic modular forms


Zbl 1207.33012
Full Text: DOI arXiv


[1] Bauer, G., Von den coefficienten der reihen von kugelfunctionen einer variablen, J. reine angew. math., 56, 101-121, (1859)
[2] Ekhad, S.B.; Zeilberger, D., A WZ proof of Ramanujan’s formula for π, (), 107-108 · Zbl 0849.33003
[3] Guillera, J., Some binomial series obtained by the WZ-method, Adv. in appl. math., 29, 4, 599-603, (2002) · Zbl 1013.33010
[4] Guillera, J., About a new kind of Ramanujan-type series, Experiment. math., 12, 4, 507-510, (2003) · Zbl 1083.33004
[5] Guillera, J., Generators of some Ramanujan formulas, Ramanujan J., 11, 1, 41-48, (2006) · Zbl 1109.33029
[6] Lehmer, E., On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Ann. of math. (2), 39, 2, 350-360, (1938) · JFM 64.0095.04
[7] McCarthy, D.; Osburn, R., A p-adic analogue of a formula of Ramanujan, Arch. math. (basel), 91, 6, 492-504, (2007) · Zbl 1175.33004
[8] Morita, Y., A p-adic analogue of the γ-function, J. fac. sci. univ. Tokyo sect. I A, 22, 255-266, (1975) · Zbl 0308.12003
[9] Morley, F., Note on the congruence \(2^{4 n} \equiv(- 1)^n(2 n)! /(n!)^2\), where \(2 n + 1\) is a prime, Ann. of math., 9, 168-170, (1895) · JFM 26.0208.02
[10] Mortenson, E., A p-adic supercongruence conjecture of Van hamme, Proc. amer. math. soc., 136, 12, 4321-4328, (2008) · Zbl 1171.11061
[11] Ramanujan, S., Modular equations and approximations to π, (), 45, 23-39, (1914), reprinted · JFM 45.1249.01
[12] Van Hamme, L., Some conjectures concerning partial sums of generalized hypergeometric series, (), 223-236 · Zbl 0895.11051
[13] Wolstenholme, J., On certain properties of prime numbers, Quart. J. math. (Oxford), 5, 35-39, (1862)
[14] Zudilin, W., Ramanujan-type formulae for \(1 / \pi\): A second wind?, (), 179-188, (2007) · Zbl 1159.11053
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.