A p-adic supercongruence for a Ramanujan-type sum

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Let p≡5(mod8)p\equiv 5\pmod{8} be a prime, and let (a)k=a(a+1)⋯(a+k−1)(a)_k=a(a+1)\cdots(a+k-1) denote the rising factorial. The p-adic supercongruence conjecture.

∑k=0p−1(12k+1)(14)k2(18)k4kk!3≡0(modp3).\sum_{k=0}^{p-1}(12 k+1) \frac{(\frac{1}{4})_{k}^{2}(\frac{1}{8})_{k}}{4^{k} k !^{3}} \equiv 0 \pmod {p^3}.

This is presented as a stronger version of a previously proved congruence modulo p2p^2, based on numerical calculations; its status is open in the source.

References

Primary source

Yudong Liu and Xiaoxia Wang, “Some q-supercongruences from Rahman's summation formula”, arXiv:2103.06416 (2021).

Additional references

7 papers in this index state this conjecture (2019–2021). The statement above is taken from the most recent of them; the others are arXiv:2009.03889, arXiv:2002.08814, arXiv:1911.01790, arXiv:1909.13173, arXiv:1907.09391, arXiv:1903.03766.

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