Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let p > 3 be a prime. We show that Tp−1 ≡ (p 3 ) 3p−1 (mod p), where the central trinomial coefficient Tn is the constant term in the expansion of (1 + x + x−1)n. We also prove three congruences modulo p conjectured by Sun, one of which is p−1 ∑ k=0 ( p− 1 k )( 2k k ) ((−1) − (−3)−k) ≡ (p 3 ) (3p−1 −...

We prove that if the signed binomial coefficient (−1) ` k i ́ viewed modulo p is a periodic function of i with period h in the range 0 ≤ i ≤ k, then k + 1 is a power of p, provided h is prime to p and not too large compared to k. (In particular, 2h ≤ k suffices.) As an application, we prove that if G and H are multiplicative subgroups of a finite field, with H < G, and such that 1− α ∈ G for all...

We prove divisibility properties for sums of powers of binomial coefficients and of q-binomial coefficients. Dedicated to the memory of Paul Erdős