On Two Congruences Involving Central Binomial Coefficients

Let p≥3 be a prime. In 2010,Z.W.Sun and R.Tauraso[12]deter-mined∑k=1p-1)(-1)k(k2k)/(kmk-1)modulo p,where m is an integer not divisible by p. Recently,Zhi-Wei Sun proposed various conjectures for further congru-ences.In particular,he made explicit conjectures on∑k=1p-1(k2k)/(k2k)medulo p3 and∑k=1p-1(-2)k(k2k)/k modulo p3.The main purpose of this thesis is to establish the following two congruences conjectured by Sun: and where qp(2):(2p-1-1)/p is the Fermat quotient with base 2 and Bp-3 is the (p-3)th Bernoulli number.A key lemma in the proof is motivated by A.Granville's work [3],and we also employ two auxiliary identities given by Z.W.Sun and R.Tauraso[12].