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 the ratio of values of a polynomial

For given positive integersa andb, the equationa(x + 1)… (x + k) =b(y+1)… (y + k) in positive integers is considered. More general equations are also considered.     

Laws of large numbers for polynomial hypergroups and some applications

There are introduced moments on polynomial hypergroups. These moments are used to prove strong laws of large number (SSLLNs) for random walks on the nonnegative integers that are homogeneous with respect to a polynomial hypergroup where SLLNs of different kind appear for polynomial hypergroups thth different properties. Furthermore, we discuss polynomial hypergroups that are associated with some discrete semigroups in a canonical way, and, using SLLNs for polynomial hypergroups, we get SLLNs for isotropic random walks on some discrete semigroups.     

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.     

Equal values of pyramidal numbers

Let ${\text{Pyr}}_m\left(x\right)$ denote the pyramidal number with integer parameters $m\ge 3$ and $x\ge 3$. In this note we investigate the Diophantine equation ${\text{Pyr}}_m\left(x\right)={\text{Pyr}}_n\left(y\right)$ in positive integer unknowns $x$ and $y$, where $m$ and $n$ are given, different integers. We deduce an effective upper bound for the size of the solutions. Our proof is based on the basic properties of elliptic curves, and elliptic integrals.     

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.     

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 induced Folkman numbers

In 1970, Folkman proved that for any graph G there exists a graph H with the same clique number as G. In addition, any r ‐coloring of the vertices of H yields a monochromatic copy of G. For a given graph G and a number of colors r let f(G, r) be the order of the smallest graph H with the above properties. In this paper, we give a relatively small upper bound on f(G, r) as a function of the order of G and its clique number. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 40, 493–500, 2012     

On Miki's identity for Bernoulli numbers

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

     

Harmonic numbers at half integer values

Half integer values of harmonic numbers and reciprocal binomial coefficients sums are investigated in this paper. Closed-form representations and integral expressions are developed for the infinite series.     

Generalized Bessel numbers and some combinatorial settings

Stirling numbers and Bessel numbers have a long history, and both have been generalized in a variety of directions. Here, we present a second level generalization that has both as special cases. This generalization often preserves the inverse relation between the first and second kind, and has simple combinatorial interpretations. We also frame the discussion in terms of the exponential Riordan group. Then the inverse relation is just the group inverse, and factoring inside the group leads to many results connecting the various Stirling and Bessel numbers.     

On arithmetic and asymptotic properties of up-down numbers

Let σ=(σ₁,…,σ_N), where σ_i=±1, and let C(σ) denote the number of permutations π of 1,2,…,N+1, whose up-down signature sign(π(i+1)-π(i))=σ_i, for i=1,…,N. We prove that the set of all up-down numbers C(σ) can be expressed by a single universal polynomial Φ, whose coefficients are products of numbers from the Taylor series of the hyperbolic tangent function. We prove that Φ is a modified exponential, and deduce some remarkable congruence properties for the set of all numbers C(σ), for fixed N. We prove a concise upper bound for C(σ), which describes the asymptotic behaviour of the up-down function C(σ) in the limit C(σ)?(N+1)!.     

Smoothed analysis of some condition numbers

In this note we do a smoothed analysis, in the sense of ( http://www‐math.mit.edu/～spielman/SmoothedAnalysis/ ), of the condition number for the Moore–Penrose inverse. Usual average analysis follows in a trivial manner as follow similar analyses for the condition number of the polar factorization. Copyright © 2005 John Wiley & Sons, Ltd.     

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.     

A classification of (some) Pisot-Cyclotomic numbers

A complete classification of degree 2, 3 and degree 4 Pisot-Cyclotomic numbers is given. Some examples of higher degrees are also given. Pisot-Cyclotomic numbers have applications to quasicrystals and quasilattices.     

Fibonacci numbers, Lucas numbers and integrals of certain Gaussian processes

We study the distributions of integrals of Gaussian processes arising as limiting distributions of test statistics proposed for treating a goodness of fit or symmetry problem. We show that the cumulants of the distributions can be expressed in terms of Fibonacci numbers and Lucas numbers.

     

Bounds on some van der Waerden numbers

For positive integers s and k₁,k₂,…,ks, the van der Waerden number w(k₁,k₂,…,ks;s) is the minimum integer n such that for every s-coloring of set {1,2,…,n}, with colors 1,2,…,s, there is a ki-term arithmetic progression of color i for some i. We give an asymptotic lower bound for w(k,m;2) for fixed m. We include a table of values of w(k,3;2) that are very close to this lower bound for m=3. We also give a lower bound for w(k,k,…,k;s) that slightly improves previously-known bounds. Upper bounds for w(k,4;2) and w(4,4,…,4;s) are also provided.     

Sums of characteristic values of compact polynomial operator pencils

We consider polynomial pencils whose coefficients are compact operators. Bounds for the sums of absolute values, real and imaginary parts of characteristic values are derived. Applications to differential and difference equations are discussed.