On congruences related to central binomial coefficients

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It is known that ∑ k = 0 ∞ ( 2 k k ) ( 2 k + 1 ) 4 k = π 2 and ∑ k = 0 ∞ ( 2 k k ) ( 2 k + 1 ) 16 k = π 3 . In this paper we obtain their p-adic analogues such as ∑ p / 2 < k < p ( 2 k k ) ( 2 k + 1 ) 4 k ≡ 3 ∑ p / 2 < k < p ( 2 k k ) ( 2 k + 1 ) 16 k ≡ p E p − 3 ( mod p 2 ) , where p > 3 is a prime and E 0 , E 1 , E 2 , … 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 = 0 p − 1 ( 2 k k ) 3 ≡ { 4 x 2 − 2 p ( mod p 2 ) if ( p 7 ) = 1 & p = x 2 + 7 y 2 ( x , y ∈ Z ) , 0 ( mod p 2 ) if ( p 7 ) = − 1 , i.e., p ≡ 3 , 5 , 6 ( mod 7 ) .