In the past few years,many congruences on sums of combinatorial numbers have been widely studied.Various supercongruences have been also conjectured by many authors,including Beukers,van Hamme,Rodriguez-Villegas,Zudilin,Z.-W.Sun,Z.-H.Sun and Guo.In this thesis,we will prove some congruences involving Delannoy number-s,Schroder numbers,Catalan numbers,Schroder-like numbers,central binomial coefficients and q-binomial coefficients,which were conjectured by Apagodu,Zeil-berger,Z.-W.Sun and Guo.For example,we prove that for any prime p ? 5 and positive integer n,which extends a result of Sun and Tauraso,and was originally conjectured by Apagodu and Zeilberger.To establish the relationship between the number of points over Fp on hyper-geometric Calabi-Yau manifolds and truncated hypergeometric series,Rodriguez-Villegas conjectured 22 interesting supercongruences for truncated hypergeomet-ric series.Some of these conjectures were gradually proved by several authors,including van Hamme,Ishikawa,Ahlgren,Mortenson,Fuselier,McCarthy,Kil-bourn and Z.-W.Sun.In this thesis,we will also generalize the Rodriguez-Villegas supercongruences for truncated hypergeometric series 2F1 and 3F2. |