## A $$p$$-adic supercongruence conjecture of van Hamme.(English)Zbl 1171.11061

The author proves the following conjecture of von Hamme: Let $$p$$ be an odd prime; then $\sum^{(p-1)/2}_{k=0} (4k+ 1){-1/2\choose k}^2\equiv p\Phi_p(- 1)\pmod{p^3},\tag{$$*$$}$ where $$\Phi_p(x)$$ is the Legendre symbol modulo $$p$$.
Formula $$(*)$$ is the $$p$$-adic analog of Ramanujan’s formula: $\sum^\infty_{k=0} (4k+1){-1/2\choose k}^3= 2/\pi.$

### MSC:

 11B65 Binomial coefficients; factorials; $$q$$-identities 11A07 Congruences; primitive roots; residue systems 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 33C20 Generalized hypergeometric series, $${}_pF_q$$ 33E50 Special functions in characteristic $$p$$ (gamma functions, etc.)
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### References:

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