Ramanujan-type supercongruences. (English) Zbl 1231.11147

Zudilin, Wadim

J. Number Theory 129, No. 8, 1848-1857 (2009).

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.

Reviewer: Florian Luca (Morelia)

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

Keywords:

supercongruence; Ramanujan series for \(1/\pi\); hypergeometric series; WZ-method

Citations:

Zbl 1207.33012

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