相关论文: On a congruence only holding for primes II

We calculate the probability that random polynomial matrices over a finite field with certain structures are right prime or left prime, respectively. In particular, we give an asymptotic formula for the probability that finitely many…

We give an equivalent form of the Twin prime conjecture relating to a symmetric property that is observed for terms present in a certain sequence of arithmetic progressions defined for a pair of co-prime integers.

Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $\Phi^*_n(q)$, which is closely related to the cyclotomic polynomial $\Phi_n(x)$ and to primitive prime…

We give a formula for computing the characteristic polynomial for certain hyperplane arrangements in terms of the number of bipartite graphs of given rank and cardinality.

In this note we show a simple formula for the coefficients of the polynomial associated with the sums of powers of the terms of an arbitrary arithmetic progression. This formula consists of a double sum involving only ordinary binomial…

We estimate the number of primes represented by a general quadratic polynomial with discriminant $\Delta$, assuming that the corresponding real character is exceptional.

We give a simple combinatorial model for plethysm. Precisely, the bialgebra dual to plethystic substitution is realised as the homotopy cardinality of the incidence bialgebra of an explicit simplicial groupoid, obtained from surjections by…

We present some congruences modulo $p^{6-d}$ for sums of the type $\sum_{k=0}^{(p-3)/2}x^k{2k\choose k}/(2k+1)^d$, for $d=1,2,3$ where $p>5$ is a prime.

This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.

We study the complexity of the model checking problem, for fixed model A, over certain fragments L of first-order logic. These are sometimes known as the expression complexities of L. We obtain various complexity classification theorems for…

Recently there has been quite a bit of study carried out related to arithmetic properties of overpartitions into non-multiples of two co-prime integers. The paper [19] by Nadji et al. looked into congruences modulo $3$ and powers of $2$ for…

In this paper we obtain some sophisticated combinatorial congruences involving binomial coefficients and confirm two conjectures of the author and Davis. They are closely related to our investigation of the periodicity of the sequence…

We study the number of real critical points of a cyclotomic polynomial $\Phi_{n}(x)$, that is, the real roots of $\Phi_{n}^{\prime}(x)$. As usual, one can, without losing generality, restrict $n$ to be the product of distinct odd primes,…

We use model theoretic techniques to construct explicit first-order axiomatizations for the classes of posets that can be represented as systems of sets, where the order relation is given by inclusion, and existing meets and joins of…

We propose a criterion that allows one to distinguish prime numbers from compound ones. This criterion is based on the counting of small quadratic residues.

The purpose of this note is to suggest an analogue for genus 2 curves of part of Gross and Zagier's work on elliptic curves. Experimentally, for genus 2 curves with CM by a quartic CM field K, it appears that primes dividing the…

We prove that certain basic hypergeometric series truncated at $k=n-1$ have the factor $\Phi_n(q)^2$, where $\Phi_n(q)$ is the $n$-th cyclotomic polynomial. This confirms two recent conjectures of the author and Zudilin. We also put forward…

Generalizing a recent proposal leading to one-parameter families of Hamiltonians and to new sets of squeezed states, we construct larger classes of physically admissible Hamiltonians permitting new developments in squeezing. Coherence is…

In the paper, we generalize some congruences of Lehmer for general composite numbers.

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…