关于Lucas数列同余性质的研究

将二项式系数的性质应用到Lucas数列的研究中,并结合Fibonacci数列与Lucas数列的恒等式得到几个有趣的Lucas数列的同余式．     

Smooth numbers and the norms of arithmetic Dirichlet convolutions

In this paper we consider the problem of exactly evaluating the p-norm of a linear operator linked with arithmetic Dirichlet convolutions. We prove that a simply derived upper bound for this norm is actually attained for several different classes of arithmetic functions including completely multiplicative functions, but not for certain multiplicative functions. Our proof depends fundamentally on the asymptotic distribution properties of smooth numbers.     

关于完全平方数的一个性质

运用不定方程的理论讨论了完全平方数的一个基本性质,得到了关于完全平方数的几个重要定理.     

On some properties of the Euler's factor of certain odd perfect numbers

Let n=π^α3^2βQ^2β be an odd positive integer, with π prime, π≡α≡1 (mod 4), Q squarefree, (Q,π)=(Q,3)=1. It is shown that: if n is perfect, then σ(π^α)≡0. Some corollaries concerning the Euler's factor of odd perfect numbers of the above mentioned form, if any, are deduced.     

Arithmetical properties of hypergeometric Bernoulli numbers

In a recent paper, Byrnes et al. (2014) have developed some recurrence relations for the hypergeometric zeta functions. Moreover, the authors made two conjectures for arithmetical properties of the denominators of the reduced fraction of the hypergeometric Bernoulli numbers. In this paper, we prove these conjectures using some recurrence relations. Furthermore, we assert that the above properties hold for both Carlitz and Howard numbers.     

On Miki's identity for Bernoulli numbers

We give a short proof of Miki's identity for Bernoulli numbers,

     

On some polynomial values of repdigit numbers

We study the equal values of repdigit numbers and the k dimensional polygonal numbers. We state some effective finiteness theorems, and for small parameter values we completely solve the corresponding equations.     

On matrices related with Fibonacci and Lucas numbers

In this paper, we obtain some new results on matrices related with Fibonacci numbers and Lucas numbers. Also, we derive the relation between Pell numbers and its companion sequence by using our representations.     

On an approximation property of Pisot numbers

Let 1<q<2 be a real number, m≥1 be a rational integer and l_m(q)={|P(q)|,P∈Z[X],P(q)≠0,H(P)≤m}, where Z[X] denotes the set of polynomials P with rational integer coefficients and H(P) is the height of P. The value of l_m(q) was determined for many particular Pisot numbers ([3] and [7]). In this paper we determine the infimum and the supremum of the numbers l_m(q) for any fixed m. We also determine the greatest limit point for the case m=1. This revised version was published online in June 2006 with corrections to the Cover Date.     

On a variant of Giuga numbers

In this paper, we characterize the odd positive integers n satisfying the congruence∑n -1 j=1 j n-1/2 ≡ 0 (mod n). We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8.     

The combinatorial properties of the Benoumhani polynomials for the Whitney numbers of Dowling lattices

In the papers (Benoumhani 1996;1997), Benoumhani defined two polynomials $F_{m,n,1}\left(x\right)$ and $F_{m,n,2}\left(x\right)$. Then, he defined $A_m(n,k)$ and $B_m(n,k)$ to be the polynomials satisfying $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{A}_m(n,k)x^{n?k}{(x+1)}^k$ and $F_{m,n,1}\left(x\right)={\sum }_{k=0}^n{B}_m(n,k)x^{n?k}{(x+1)}^k$. In this paper, we give a combinatorial interpretation of the coefficients of $A_{m+1}(n,k)$ and prove a symmetry of the coefficients, i.e., $\left[m^s\right]A_{m+1}(n,k)=\left[m^{n?s}\right]A_{m+1}(n,n?k)$. We give a combinatorial interpretation of $B_{m+1}(n,k)$ and prove that $B_{m+1}(n,n?1)$ is a polynomial in $m$ with non-negative integer coefficients. We also prove that if $n\ge 6$ then all coefficients of $B_{m+1}(n,n?2)$ except the coefficient of $m^{n?1}$ are non-negative integers. For all $n$, the coefficient of $m^{n?1}$ in $B_{m+1}(n,n?2)$ is $?(n?1)$, and when $n\le 5$ some other coefficients of $B_{m+1}(n,n?2)$ are also negative.     

On graphs with small Ramsey numbers*

Let R(G) denote the minimum integer N such that for every bicoloring of the edges of K_N, at least one of the monochromatic subgraphs contains G as a subgraph. We show that for every positive integer d and each γ,0 < γ < 1, there exists k = k(d,γ) such that for every bipartite graph G = (W,U;E) with the maximum degree of vertices in W at most d and , . This answers a question of Trotter. We give also a weaker bound on the Ramsey numbers of graphs whose set of vertices of degree at least d + 1 is independent. © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 198–204, 2001     

On some Rado numbers for generalized arithmetic progressions

The 2-color Rado number for the equation x₁+x₂−2x₃=c, which for each constant we denote by S₁(c), is the least integer, if it exists, such that every 2-coloring, Δ : [1,S₁(c)]→{0,1}, of the natural numbers admits a monochromatic solution to x₁+x₂−2x₃=c, and otherwise S₁(c)=∞. We determine the 2-color Rado number for the equation x₁+x₂−2x₃=c, when additional inequality restraints on the variables are added. In particular, the case where we require x₂<x₃<x₁, is a generalization of the 3-term arithmetic progression; and the work done here improves previously established upper bounds to an exact value.     

On the combinatorics of Galois numbers

We define interval decompositions of the lattice of subspaces of a finite-dimensional vector space. We show that such a decomposition exists if and only if there exists a family of linear forms with certain properties. As applications we prove that all finite-dimensional real vector spaces admit an interval decomposition, while GF(2)ⁿ has an interval decomposition if and only if n≤4. On the other hand, we present an interval decomposition of GF(3)⁵. This partially answers a question of Faigle and Kruse (2004) [1] and [4].     

On some three color Ramsey numbers for paths and cycles

For given graphs G₁,G₂,…,G_k, k≥2, the multicolor Ramsey number, denoted by R(G₁,G₂,…,G_k), is the smallest integer n such that if we arbitrarily color the edges of a complete graph on n vertices with k colors, there is always a monochromatic copy of G_i colored with i, for some 1≤i≤k. Let P_k (resp. C_k) be the path (resp. cycle) on k vertices. In the paper we consider the value for numbers of type R(P_i,P_k,C_m) for odd m, k≥m≥3 and when i is odd, and when i is even. In addition, we provide the exact values for Ramsey numbers R(P₃,P_k,C₄) for all integers k≥3.     

On a variance associated with the distribution of primes in arithmetic progressions

As is usual in prime number theory, write It is well known that when q is close to x the averagevalue of is about xlog q,and recently Friedlander and Goldston have shown that if then the first moment of V(x,q)-U(x,q)is small. In this memoir it is shown that the same is true forall moments. 2000 Mathematics Subject Classification: 11N13.     

On the maximum of r-Stirling numbers

Determining the location of the maximum of Stirling numbers is a well-developed area. In this paper we give the same results for the so-called r-Stirling numbers which are natural generalizations of Stirling numbers.     

On computing the distinguishing and distinguishing chromatic numbers of interval graphs and other results

A vertex k-coloring of graph G is distinguishing if the only automorphism of G that preserves the colors is the identity map. It is proper-distinguishing if the coloring is both proper and distinguishing. The distinguishing number ofG, D(G), is the smallest integer k so that G has a distinguishing k-coloring; the distinguishing chromatic number ofG, χ_D(G), is defined similarly.It has been shown recently that the distinguishing number of a planar graph can be determined efficiently by counting a related parameter-the number of inequivalent distinguishing colorings of the graph. In this paper, we demonstrate that the same technique can be used to compute the distinguishing number and the distinguishing chromatic number of an interval graph. We make use of PQ-trees, a classic data structure that has been used to recognize and test the isomorphism of interval graphs; our algorithms run in O(n³log³n) time for graphs with n vertices. We also prove a number of results regarding the computational complexity of determining a graph’s distinguishing chromatic number.