Three pairs of congruences concerning sums of central binomial coefficients.

Recently the first author proved a congruence proposed in 2006 by Adamchuk: ∑ k = 1 ⌊ 2 p / 3 ⌋ 2 k k ≡ 0 (mod p 2) for any prime p = 1 (mod 3). In this paper, we provide more examples (with proofs) of congruences of the same kind ∑ k = 1 ⌊ a p / r ⌋ 2 k k x k (mod p 2) where p is a prime such that p ≡ 1 (mod r) , a / r is a fraction in (1 / 2 , 1) and x is a p -adic integer. The key ingredients are the p -adic Gamma function Γ p and a special class of computer-discovered hypergeometric identities. [ABSTRACT FROM AUTHOR]