Congruences for the second-order Catalan numbers

Abstract

Abstract. Let p be any odd prime. We mainly show thatis the kth Catalan number of order 2.

Factors of Sums and Alternating Sums Involving Binomial Coefficients and Powers of Integers

“…In this section we shall use the following theorem of Lucas (see, for example, [7]). We refer the reader to [10,14,16] for recent applications of Lucas' theorem.…”

Some q-congruences related to 3-adic valuations

“…where (−) denotes the Legendre symbol. (See also [ST2,ZPS,S09a,S09b,S09c] for related results.) When checking whether there are composite numbers n such that…”

Cubic congruences and sums involving 3k k

“…Let Z be the set of integers, and for a prime p let Z p denote the set of those rational numbers whose denominator is not divisible by p. Let p > 5 be a prime. In [12] Zhao, Pan and Sun proved that p−1 k=1 3k k 2 k ≡ 6 5 ((−1) (p−1)/2 − 1) (mod p). 1 The author is supported by the Natural Sciences Foundation of China (grant no.…”