Three pairs of congruences concerning sums of central binomial coefficients

Recently the first author proved a congruence proposed in 2006 by Adamchuk: Sigma([2p/3])(k=1) (2k 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 Sigma([ap/r])(k=1) (2k 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 Gamma(p) and a special class of computer-discovered hypergeometric identities.