Some Super Congruences Involving Binomial Coefficients

Abstract

Let p > 3 be a prime. We show that T p−1 ≡ p 3 3 p−1 (mod p 2), where the central trinomial coefficient T n is the constant term in the expansion of (1+x+x −1) n. We also prove three congruences conjectured by Sun one of which is as follows: p−1 k=0 p − 1 k 2k k ((−1) k − (−3) −k) ≡ p 3 (3 p−1 − 1) (mod p 3). 

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