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We consider two kinds of congruence proposed recently by Sun. The first one is of the form 1 n ( a p n − χ p ⋅ a n ) ≡ 0 ( mod p 2 ) , where p is a prime, χ p is a character and { a n } n ≥ 1 is a sequence of integers. By using a constant term expression for 2 k k , we derive congruence for two sequences { a n } n ≥ 1 . The second one involves the sum S ( a , x ) = ∑ k = 0 p − 1 a k d k ( x ) , where d n ( x ) = ∑ k = 0 n n k x k 2 k. By giving a constant term expression for d n ( x ) , we derive a congruence for S ( a , x ) modulo a prime p. [ABSTRACT FROM AUTHOR]

The Delannoy numbers and Schröder numbers are given by D n = ∑ k = 0 n n k n + k k and S n = ∑ k = 0 n n k n + k k 1 k + 1 , respectively. We mainly prove that for any prime p ≥ 5 , ∑ k = 1 ( p − 1 ) / 2 D k S k ≡ 4 x 2 ( mod p ) if  p ≡ 1 ( mod 4 ) and p = x 2 + 4 y 2 , 0 ( mod p ) if  p ≡ 3 ( mod 4 ) , which was originally conjectured by Sun in 2011. A related congruence on the Delannoy numbers is also proved. [ABSTRACT FROM AUTHOR]

Let be the Franel numbers given by , and let be a prime. In this paper, we mainly determine for . [ABSTRACT FROM AUTHOR]

Let be a prime greater than and let be a rational -adic integer. In this paper, we try to determine , and reveal the connection between cubic congruences and the sum , where is the greatest integer not exceeding . Suppose that are rational -adic integers, , and . In this paper, we show that the number of solutions of the congruence depends only on . Let be a prime of the form and so with . When and , we establish congruences for and modulo p. As a consequence, when we show that has three solutions if and only if is a cubic residue of . [ABSTRACT FROM AUTHOR]

The notion of Fu's k dots bracelet partitions was introduced by Shishuo Fu. For any positive integer k, let 픅k(n) denote the number of Fu's k dots bracelet partitions of n. Fu also proved several congruences modulo primes and modulo powers of 2. Recently, Radu and Sellers extended the set of congruences proven by Fu by proving three congruences modulo squares of primes for 픅5(n), 픅7(n) and 픅11(n). More recently, Cui and Gu, and Xia and the author derived a number of congruences modulo powers of 2 for 픅5(n). In this paper, we prove four congruences modulo 2 and two congruences modulo 4 for 픅9(n) by establishing the generating functions of 픅9(An+B) modulo 4 for some values of A and B. [ABSTRACT FROM AUTHOR]

Let p be a prime greater than 3. In this paper, by using expansions and congruences for Lucas sequences and the theory of cubic residues and cubic congruences, we establish some congruences for and modulo p, where [x] is the greatest integer not exceeding x, and m is a rational p-adic integer with m ≢ 0 ( p). [ABSTRACT FROM AUTHOR]

Suppose that p is an odd prime and d is a positive integer. Let x and y be integers given by p = x2 + dy2 or 4p = x2 + dy2. In this paper we determine x (mod p) for many values of d. For example,... where x is chosen so that x = 1 (mod 3). We also pose some conjectures on supercongruences modulo p concerning binary quadratic forms [ABSTRACT FROM AUTHOR]