Related papers: Congruence properties for the trinomial coefficien…
We prove several congruences for trinomial coefficients.
In this paper, we gave some properties of binomial coefficient.
Using generalized binomial coefficients with respect to fundamental Lucas sequences we establish congruences that generalize the classical congruence of Wolstenholme and other related stronger congruences.
We describe a congruence property of solvable polynomials over Q, based on the irreducibility of cyclotomic polynomials over number fields that meet certain conditions.
We provide elementary proof of several congruences involving single sum and multisums of binomial coefficients.
In this article, we derive a congruence property of particular sum rules involving prime numbers. The resulting expression involves Bernoulli numbers and polynomials, for which we obtain, as a consequence, a general congruence relation as…
In this paper one extends the binomial and trinomial coefficients to the concept of 'k-nomial' coefficients, and one obtains some properties of these. As an application one generalizes Pascal's triangle.
In this paper, we describe a congruence property of solvable polynomials with coefficients in the Gaussian field Q(i).
In this paper, we establish congruences (mod $p^2$) involving the quadrinomial coefficients $\dbinom{np-1}{p-1}_{3}$ and $\dbinom{np-1}{\frac{p-1}{2}}_{3}$. This is an analogue of congruences involving the trinomial coefficients…
The aim of this work is to establish congruences $\left( \operatorname{mod}p^{2}\right) $ involving the trinomial coefficients $\binom{np-1}{p-1}_{2}$ and $\binom{np-1}{\left( p-1\right)/2}_{2}$ arising from the expansion of the powers of…
In this paper, we present several new congruences on the $q$-trinomial coefficients introduced by Andrews and Baxter. A new congruence on sums of central $q$-binomial coefficients is also established.
We prove that the Eulerian polynomial satisfies certain polynomial congruences. Furthermore, these congruences characterize the Eulerian polynomial.
We give elementary proofs of some congruence criteria to compute binomial coefficients in modulo a prime. These criteria are analogues to the symmetry property of binomial coefficients. We give extended version of Lucas Theorem by using…
In this paper, we pose many challenging conjectures on congruences involving binomial coefficients and Ap\'ery-like numbers.
In this paper, with the help of trinomial coefficients we study some arithmetic properties of certain determiants involving reciprocals of binary quadratic forms over finite fields.
We present a detailed proof of Wolstenholme's theorem using an Egorychev-type contour integral and an exponential change of variables. All formal series manipulations are justified, and the connection with harmonic sums and Bernoulli…
Elementary proofs of Sylvester's, Wolstenholme's, Morley's and Lehmer's congruence theorems
In this paper, we will present several new congruences involving binomial coefficients under integer moduli, which are the continuation of the previous two work by Cai \textit{et al.} (2002, 2007).
We establish a congruence on sums of central $q$-binomial coefficients. From this $q$-congruence, we derive the divisibility of the $q$-trinomial coefficients introduced by Andrews and Baxter.
We give an elementary and self-contained proof of the equivalence of a collection of Wolstenholme-type congruences due to Helou and Terjanian.