Search

We show that integral solutions of x 3 + y 3 + z 3 = q , where q is prime, satisfy certain congruence conditions. [ABSTRACT FROM AUTHOR]

Let a , n , m ∈ ℤ + with (a , p) = 1 and p be an odd prime. We find a supercongruence for ∑ p | k a p n k m and related sums of powers of binomial coefficients. These results complement prior results for ∑ k ≡ i (mod   p) a p n k m with i = 1 , 2 , … , p − 1 obtained recently by the author. [ABSTRACT FROM AUTHOR]

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]

New families of even non-congruent numbers with arbitrarily many distinct prime factors are presented. Methods for generating even non-congruent numbers from existing non-congruent numbers are also given. These results yield new even non-congruent numbers whose factorizations contain primes in all four odd congruence classes modulo eight. [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]

Wang and Cai established the following harmonic congruence for odd prime and positive integer : where denote the set of positive integers which are prime to . In this paper, the authors generalize it into super congruence for odd prime and positive integer : where and are positive integers such that [ABSTRACT FROM AUTHOR]

In 2014, Wang and Cai established the following harmonic congruence for any odd prime and positive integer , where and denote the set of positive integers which are prime to . In this paper, we obtain an unexpected congruence for distinct odd primes , and positive integers , and the necessary and sufficient condition for Finally, we raise a conjecture that for and odd prime power , , However, we fail to prove it even for with three distinct prime factors. [ABSTRACT FROM AUTHOR]

In recent years, the congruence first discovered by the last author has been generalized by either increasing the number of indices and considering the corresponding super congruences, or by considering the alternating version of multiple harmonic sums. In this paper, we prove a family of similar super congruences modulo prime powers with the indices summing up to where is coprime to , and where all the indices are also coprime to . [ABSTRACT FROM AUTHOR]

In this paper, we prove that the constant terms of powers of a Laurent polynomial satisfy certain congruences modulo prime powers. As a corollary, the generating series of these numbers considered as a function of a -adic variable admits an analytic continuation in a non-trivial way, as it was shown previously by Dwork for a class of hypergeometric series. [ABSTRACT FROM AUTHOR]

The harmonic numbers play important roles in mathematics. Let be a prime. With the help of some combinatorial identities, we establish the following two new congruences: and where denotes the Bernoulli polynomial of degree . As an application, we determine and modulo , where with . [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 and Q be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: U0 = 0, U1 = 1, and Un+1 = PUn + QUn-1 for n ≥ 1 and V0 = 2, V1 = P, and Vn+1 = PVn + QVn-1 for n ≥ 1, respectively. For all odd relatively prime values of P and Q such that P ≥ 1, we determine all indices n and m such that Vn = wVmx2 or VnVm = wx2 with w = 1, 2, 3 or 6 under the assumptions P2 + 4Q > 0 and Vm ≠ 1 for all positive integers m. [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]

For any prime p, we establish congruences modulo pn+1 for the difference of the pn+1th and pnth Motzkin numbers and determine the p-adic order of the difference. The results confirm recent conjectures on the order. The applied techniques involve the use of congruences for the differences of certain Catalan numbers and binomial coefficients, congruential identities for sums of Catalan numbers, central binomial and trinomial coefficients, infinite incongruent disjoint covering systems and the solution of congruential recurrences. [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]

In 2007, Andrews and Paule introduced the family of functions ΔK(n) which enumerate the number of broken fc-diamond partitions for a fixed positive integer k. In that paper, Andrews and Paule proved that, for all n ≥ 0, Δ1(2n + 1) ≡ 0 (mod 3) using a standard generating function argument. Soon after, Shishuo Fu provided a combinatorial proof of this same congruence. Fu also utilized this combinatorial approach to naturally define a generalization of broken k-diamond partitions which he called k dots bracelet partitions. He denoted the number of k dots bracelet partitions of n by Bk(n) and proved various congruence properties for these functions modulo primes and modulo powers of 2. In this note, we extend the set of congruences proven by Fu by proving the following congruences: For all n ≥ 0, B5(10n + 7)≡0 (mod 5²), B7(14n+ 11)≡ 0 (mod 7²), and B11(22n + 21)≡ 0 (mod ll²) We also conjecture an infinite family of congruences modulo powers of 7 which are satisfied by the function B7. [ABSTRACT FROM AUTHOR]

Let n ≥ 3 be an integer. For any permutation o of A = {0,1,…, n -- 1}, set (σ, n) = …· It is proved that n is prime if and only if (σ, n) = 0 (mod n) for any σ. When n is composite, we show that there exist cycles of any length satisfying this congruence. We show as well that the analog statement is true for cycles of length at least 2 not satisfying the congruence. [ABSTRACT FROM AUTHOR]

Let L/K be a Galois extension of number fields. The problem of counting the number of prime ideals 픭 of K with fixed Frobenius class in Gal(L/K) and norm satisfying a congruence condition is considered. We show that the square of the error term arising from the Chebotarëv Density Theorem for this problem is small "on average". The result may be viewed as a variation on the classical Barban-Davenport-Halberstam Theorem. [ABSTRACT FROM AUTHOR]

Let Q(x, y) be a quadratic form with discriminant D ≠ 0. We obtain non-trivial upper bound estimates for the number of solutions of the congruence Q(x, y) ≡ λ (mod p), where p is a prime and x, y lie in certain intervals of length M, under the assumption that Q(x, y) - λ is an absolutely irreducible polynomial modulo p. In particular, we prove that the number of solutions to this congruence is Mo(1) when M ≪ p1/4. These estimates generalize a previous result by Cilleruelo and Garaev on the particular congruence xy ≡ λ(mod p). [ABSTRACT FROM AUTHOR]

Let hn denote the class number of ${\mathbb Q}(2\cos\frac{2\pi}{2^{n+2}})$ which is a cyclic extension of degree 2n over the rational number field ℚ. There are no known examples of hn > 1. We prove that a prime number ℓ does not divide hn for all n ≥ 1 if ℓ is less than 109 or ℓ satisfies a congruence relation ℓ ≢ ± 1 (mod 32). [ABSTRACT FROM AUTHOR]