On congruences related to central binomial coefficients

It is known that∑k=0∞(2kk)(2k+1)4k=π2and∑k=0∞(2kk)(2k+1)16k=π3. In this paper we obtain their p-adic analogues such as∑p/23p>3 is a prime and E0,E1,E2,…E0,E1,E2,… are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have∑k=0p−1(2kk)3≡{4x2−2p(mod p2)if(p7)=1&p=x2+7y2(x,y∈Z),0(modp2)if(p7)=−1,i.e.,p≡3,5,6(mod7).