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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.)
Citations:
Zbl 0895.11051
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References:
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