On Some New Congruences for Binomial Coefficients

Abstract

In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let p be a prime and let a be any positive integer. We determine p a −1 k=0 2k k+d mod p 2 for d = 0, . . . , p a and p a −1 k=0 2k k+δ mod p 3 for δ = 0, 1. We also show that 1 C n p a −1 k=0 C p a n+k ≡ 1 − 3(n + 1) p a − 1 3 (mod p 2 ) for every n = 0, 1, 2, . . . , where C m is the Catalan number 2m m /(m + 1), and ( · 3 ) is the Legendre symbol.

Super congruences and Euler numbers

“…[PS], [ST1] and [ST2]), Z. W. Sun [S10b] and conjectured that there are no odd composite numbers n > 1 satisfying the congruence n−1 k=0 2k k /2 k ≡ (−1) (n−1)/2 (mod n 2 ). He also searched those exceptional primes p such that p−1 k=0 2k k /2 k ≡ (−1) (p−1)/2 (mod p 3 ) and only found two such primes: 149 and 241.…”

Congruences for the second-order Catalan numbers

“…Recently Z. W. Sun and R. Tauraso [ST1,ST2] obtained some further congruences concerning sums involving Catalan numbers.…”

Congruences on sums of q-binomial coefficients

“…In combinatorics, the numbers 2k k are called central binomial coefficients. In 2011, Sun and Tauraso [10] proved that for any prime p ≥ 5,…”