## On the (K.2) supercongruence of Van Hamme.(English)Zbl 1400.11062

The authors prove the last remaining case of the original 13 Ramanujan-type supercongruence conjectures due to L. Van Hamme from 1997 [Lect. Notes Pure Appl. Math. 192, 223–236 (1997; Zbl 0895.11051)]: Let $$p$$ be an odd prime. Then
$\sum_{n=0}^{\frac{p-1}{2}} \frac{(\tfrac12)_n^3}{n!^3} (42n+5)\frac 1{64^n} \equiv 5p(-1)^{\frac{p-1}{2}}\pmod{p^4}. \tag{\text{Entry\;K.2}}$
The proof utilizes classical congruences and a WZ pair due to Guillera. Some future directions concerning this type of supercongruence are mentioned.

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 11A07 Congruences; primitive roots; residue systems 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33F10 Symbolic computation of special functions (Gosper and Zeilberger algorithms, etc.)

### Keywords:

supercongruence; Ramanujan; Wilf-Zeilberger pair

Zbl 0895.11051
Full Text:

### References:

 [1] Berndt, B., Ramanujan’s notebooks. part IV, (1994), Springer-Verlag New York · Zbl 0785.11001 [2] Borwein, J.; Borwein, P., Pi and the AGM: a study in analytic number theory and computational complexity, Canad. Math. Soc. Ser. Monogr. Adv. Texts, Wiley-Interscience Publ., (1987), John Wiley & Sons, Inc. New York · Zbl 0611.10001 [3] Greene, J., Hypergeometric functions over finite fields, Trans. Amer. Math. Soc., 301, 1, 77-101, (1987) · Zbl 0629.12017 [4] Guillera, J., Generators of some Ramanujan formulas, Ramanujan J., 11, 1, 41-48, (2006) · Zbl 1109.33029 [5] Kilbourn, T., An extension of the apéry number supercongruence, Acta Arith., 123, 335-348, (2006) · Zbl 1170.11008 [6] Long, L., Hypergeometric evaluation identities and supercongruences, Pacific J. Math., 249, 2, 405-418, (2011) · Zbl 1215.33002 [7] Long, L.; Ramakrishna, R., Some supercongruences occurring in truncated hypergeometric series, preprint available at: · Zbl 1336.33018 [8] McCarthy, D.; Osburn, R., A p-adic analogue of a formula of Ramanujan, Arch. Math. (Basel), 91, 6, 492-504, (2008) · Zbl 1175.33004 [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, Quart. J. Math., 45, 350-372, (1914) · JFM 45.1249.01 [12] Swisher, H., On the supercongruence conjectures of Van hamme, Res. Math. Sci., (2015), in press, preprint available at: · Zbl 1337.33005 [13] Van Hamme, L., Some conjectures concerning partial sums of generalized hypergeometric series, (p-Adic Functional Analysis, Nijmegen, 1996, Lecture Notes in Pure and Appl. Math., vol. 192, (1997), Dekker), 223-236 · Zbl 0895.11051 [14] Wolstenholme, J., On certain properties of prime numbers, Quart. J. Math. (Oxford), 5, 35-39, (1862) [15] Zudilin, W., Ramanujan-type formulae for $$1 / \pi$$: a second wind?, (Modular Forms and String Duality, Fields Inst. Commun., vol. 54, (2008), Amer. Math. Soc. Providence, RI), 179-188 · Zbl 1159.11053 [16] Zudilin, W., Ramanujan-type supercongruences, J. Number Theory, 129, 8, 1848-1857, (2009) · Zbl 1231.11147
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.