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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.

MSC:
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
Citations:
Zbl 1207.33012
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References:
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