Related papers: A method for determining the mod-$2^k$ behaviour o…
In Graph Theory a number of results were devoted to studying the computational complexity of the number modulo 2 of a graph's edge set decompositions of various kinds, first of all including its Hamiltonian decompositions, as well as the…
We prove some polynomial identities from which we deduce congruences modulo $p^2$ for the Fermat quotient $\frac{2^p-2}{p}$ for any odd prime $p$ (Proposition 1 and Theorem 1). These congruences are simpler than the one obtained by…
Congruence families, i.e., $\ell$-adic convergence for well-defined arithmetic subsequences, is a commonplace phenomenon for the coefficients of modular forms. Such families superficially resemble one another, but they often vary…
We determine the roots in F_{q^3} of the polynomial X^{2q^k+1} + X + c for each positive integer k and each c in F_q, where q is a power of 2. We introduce a new approach for this type of question, and we obtain results which are more…
While there has been some progress on the decomposition of Kronecker products of characters of the symmetric groups in recent times, results on the symmetric and alternating part of Kronecker squares are still scarce. Here, new results (and…
In this paper, a computationally simple and explicit method of constructing recursive sequence of primitive polynomials of degree $n2^k (k = 1, 2, 3,\ldots)$ over $\mathbb{F}_{q}$ is given.
We give a simple matrix-based proof of congruence equations modulo a prime $p$ involving sums of binomial coefficients appearing in Pascal's triangle. These equations can be used to construct some groups of exponent $p^n$. These groups, as…
Given a prime number $p$, the study of divisibility properties of a sequence $c(n)$ has two contending approaches: $p$-adic valuations and superconcongruences. The former searches for the highest power of $p$ dividing $c(n)$, for each $n$;…
We develop a new method to compute the homology groups of finite topological spaces (or equivalently of finite partially ordered sets) by means of spectral sequences giving a complete and simple description of the corresponding…
We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…
The following congruence for power sums, $S_n(p)$, is well known and has many applications: $1^n+2^n +\dots +p^n \equiv\begin{cases} -1 \text{ mod } p, & \text{ if } \ p-1 \ | \ n; 0 \text{ mod } p, & \text{ if } \ p-1 \ \not| \ n,…
We give congruences modulo powers of $p \in \{3, 5,7\}$ for the Fourier coefficients of certain modular functions in level $p$ with poles only at 0, answering a question posed by Andersen and the first author and continuing work done by the…
The harmonic numbers $H_n=\sum_{0<k\ls n}1/k\ (n=0,1,2,\ldots)$ play important roles in mathematics. With helps of some combinatorial identities, we establish the following two congruences:…
We utilize the Wilf-Zeilberger (WZ) method to establish congruences related to truncated Ramanujan-type series. By constructing hypergeometric terms $f(k, a, b, \ldots)$ with Gosper-summable differences and selecting appropriate parameters,…
The nilpotence order of the mod 2 Hecke operators. Let $\Delta=\sum_{m=0}^\infty q^{(2m+1)^2} \in F_2[[q]]$ be the reduction mod 2 of the $\Delta$ series. A modular form f modulo 2 of level 1 is a polynomial in $\Delta$. If p is an odd…
We study congruences of the form F(j(z)) | U(p) = G(j(z)) mod p, where U(p) is the p-th Hecke operator, j is the basic modular invariant 1/q+744+196884q+... for SL2(Z), and F,G are polynomials with integer coefficients. Using the interplay…
We study the space of period polynomials associated with modular forms of integral weight for finite index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner…
We study the permutation action of a finite symplectic group of characteristic 2 on the set of subspaces of its standard module which are either totally isotropic or else complementary to totally isotropic subspaces with respect to the…
In this paper we study special bases of certain spaces of half-integral weight weakly holomorphic modular forms. We establish a criterion for the integrality of Fourier coefficients of such bases. By using recursive relations between Hecke…
Inspired by the work of S. Ramanujan, many people have studied generalized modular equations and the numerous identities found by Ramanujan. These identities known as modular equations can be transformed into polynomial equations. There is…