Supercongruences concerning lacunary sums of Catalan numbers and binomial coefficients

We consider two conjectures of Sun in [22] concerning lacunary sums of Catalan numbers and binomial coefficients. As a conclusion, we completely confirm one of Sun's conjecture and partially confirm the other one. For example, suppose that p >= 3 is prime and 0 <= r <= p - 1. Then, for any a >= 0, we have S-r(p(a+2)) = S-r(p(a)) (mod p(1+a)), where S-r(p(a)) = Sigma(0<k<pa) (k equivalent to r (mod p-1)) C-k, and C-k = ((k) (2k))/(k + 1) is the k-th Catalan number. Furthermore, when p = 2, we have S-r(2(a+2)) = S-r(2(a)) (mod 2(2(1+a))).