Recent Advances for Ramanujan Type Supercongruences

In 1914, Ramanujan listed 17 infinite series representations of 1/pi of the form (k=0)Sigma(infinity) A(k)x(k) = delta/pi, which were later used by J. Borwein and P. Borwein and D. Chudnovsky and G. Chudnovsky to find approximations for pi. Several of these formulas relate hypergeometric series to values of the gamma function. In 1997, van Hamme developed a p-adic analogue of these series called Ramanujan type supercongruences and conjectured 13 formulas relating truncated sums of hypergeometric series to values of the p-adic gamma function, three of which he proved. Since then, a handful more have been proved. In this survey, we discuss various methods to prove these supercongruences, including recent geometric interpretations.