Congruences Involving Apery Polynomials And Central Binomial Coefficients

The central binomial coefficients(?)(n=0,1,2,…)play important roles in number theory and combinatorics.We show that for any odd prime p we have(?),where(÷)denotes the Legendre symbol,and the first congruence was conjectured by M.Apagodu.We also confirm the following conjecture of Z.-W.Sun:for any odd prime p,we have(?) where(?),and the Euler numbers E0,E1,E2,…are given by(?) The Apery numbers(?)(n=0,1,2,…)arose from Apery's proof of the irrationality of(?).Z.-W.Sun introduced the Apery polynomials(?) In this thesis,we also prove that:for any prime p>3,we have(1)(?);(2)(?);(3)(?).which were conjectured or observed by Z.-W.Sun.