A generalization of the formula for the triangular number of the sum and product of natural numbers

This note generalizes the formula for the triangular number of the sum and product of two natural numbers to similar results for the triangular number of the sum and product of r natural numbers. The formula is applied to derive formula for the sum of an odd and an even number of consecutive triangular numbers.     

On the largest prime divisor of an odd harmonic number

A positive integer is called a (Ore's) harmonic number if its positive divisors have integral harmonic mean. Ore conjectured that every harmonic number greater than is even. If Ore's conjecture is true, there exist no odd perfect numbers. In this paper, we prove that every odd harmonic number greater than must be divisible by a prime greater than .

     

关于高阶Genocchi数的一些恒等式

讨论了高阶Genocchi数的性质,建立了一些包含高阶Genocchi数和高阶Euler-Bernoulli数的恒等式.     

On Miki's identity for Bernoulli numbers

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

     

Prime and composite numbers as integer parts of powers

We study prime and composite numbers in the sequence of integer parts of powers of a fixed real number. We first prove a result which implies that there is a transcendental number ξ>1 for which the numbers [ξ^{n !}], n =2,3, ..., are all prime. Then, following an idea of Huxley who did it for cubics, we construct Pisot numbers of arbitrary degree such that all integer parts of their powers are composite. Finally, we give an example of an explicit transcendental number ζ (obtained as the limit of a certain recurrent sequence) for which the sequence [ζⁿ], n =1,2,..., has infinitely many elements in an arbitrary integer arithmetical progression. This revised version was published online in June 2006 with corrections to the Cover Date.     

Introduction to Fibonacci and Lucas hybrinomials

ABSTRACT

The hybrid numbers are generalization of complex, hyperbolic and dual numbers. In this paper, we introduce and study the Fibonacci and Lucas hybrinomials, i.e. polynomials, which are a generalization of the Fibonacci hybrid numbers and the Lucas hybrid numbers, respectively.     

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)!.     

Similarity of hybrid numbers

The set of hybrid numbers $𝕂$ is a noncommutative number system that unified and generalized the complex, dual, and double (hyperbolic) numbers with the relation ih =− hi =ε+ i. Two hybrid numbers p and q are said to be similar if there exist a nonlightlike hybrid number x satisfying the equality x ⁻¹ qx = p . And, it is denoted by p ∼ q . In this paper, we study the concept of similarity for hybrid numbers by solving the linear equations px = xq and qx − xp = c for $\text{◂,▸}\mathbf{p},\mathbf{q},\mathbf{c}\in 𝕂\mathbf{.}$     

Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials

The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on ? _p . From these symmetry, we can derive many interesting recurrence identities for Bernoulli and Euler polynomials. Finally we introduce the new concept of symmetry of fermionic p-adic invariant integral on ? _p . By using this symmetry of fermionic p-adic invariant integral on ? _p , we will give some relations of symmetry between the power sum polynomials and Euler numbers. The relation between the q-Bernoulli polynomials and q-Dedekind type sums which discussed in Y. Simsek (q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), pp. 333–351) can be also derived by using the properties of symmetry of fermionic p-adic integral on ? _p .     

The (p,q) property in families of d-intervals and d-trees

Given integers $p\ge q>1$, a family of sets satisfies the $(p,q)$ property if among any $p$ members of it some $q$ intersect. We prove that for any fixed integer constants $p\ge q>1$, a family of $d$-intervals satisfying the $(p,q)$ property can be pierced by $O\left(d^{\frac{q}{q-1}}\right)$ points, with constants depending only on $p$ and $q$. This extends results of Tardos, Kaiser and Alon for the case $q=2$, and of Kaiser and Rabinovich for the case $p=q=\lceil lo{g}_2(d+2)\rceil$. We further show that similar bounds hold in families of subgraphs of a tree or a graph of bounded tree-width, each consisting of at most $d$ connected components, extending results of Alon for the case $q=2$. Finally, we prove an upper bound of $O\left(d^{\frac{1}{p-1}}\right)$ on the fractional piercing number in families of $d$-intervals satisfying the $(p,p)$ property, and show that this bound is asymptotically sharp.     

Odd perfect numbers have a prime factor exceeding

Jenkins in 2003 showed that every odd perfect number is divisible by a prime exceeding . Using the properties of cyclotomic polynomials, we improve this result to show that every perfect number is divisible by a prime exceeding .

     

2500年研究探寻相亲数

设σ(n)为n的所有正因子(包括1和n本身在内)之和.正整数对(m,n)被称之为相亲数(或双亲数,因为这种数总是成双成对出现的)如果他们满足 σ(m)=σ(n) = m + n.如果n=n, σ(m)=2m,则m被称之为完全数(或单亲数,因为这种数总是单独出现的).更一般的,如果κ个(κ＞2)正整数(m1,m2,…mmk)满足下列条件σ(m1)=m1+m2,σ(m2)=m2+m3,σ(mk)=mκ+m1.则这κ个正整数被称之为多亲数.第一对相亲数(220,284)是在2500年前的古希腊数学家毕达哥拉斯发现的.不过迄今为止,人们对相亲数的情况、尤其对相亲数的分布情况仍然知之甚少.与相亲数有关的难题、尤其是悬而未决千百年的难题还很多就是在今夭,我们仍然不知道是不是有无穷多对相亲数,我们甚至连一个生成相亲数的充分必要条件(定义除外)都没有.在这篇文章中,我们试图给出人类在2500年的漫长历史长河中研究、探寻相亲数的大致情况与重要结果,并着重介绍从古至今生成相亲数的各种数值方法与代数方法.完全数的研究探寻史几乎与相亲数的研究探寻史是一样长的.比如2350年前的古希腊数学家欧几理德就在其数学名著<几何原本>中列出了前四个完全数,不过迄今为止,人们总共也只找到39个完全数,并且这些完全数还都是偶完全数.至于有没有奇完全数的存在,则是一个悬而未决两千多年的著名数学难题.最早的两串多亲数(一串为5个.另一串为28个),则是由法国数学家Poulet于1918年发现的.多亲数的研究探寻史虽然比相亲数的研究探寻史要短得多,但目前人们对它们的注意力与日俱增.由于相亲数与完全数及多亲数密切相关、紧密相连(我们可以将其统一称之为亲和数,因为它们都与相关数的因子和有关),因此在本文中,我们除了要讨论介绍相亲数外,也将顺便介绍完全数与多亲数的研究与探寻简史、以及人们在研究探寻这些数时所获得的一些重要结果.附注截止2004年3月25日作者校勘清样时,人们已经发现了共40个完全数和6262871对相亲数.     

Riordan阵与含有Genocchi数的一些恒等式

In this paper, using generating functions and Riordan arrays, we get some identities relating Genocchi numbers with Stirling numbers and Cauchy numbers.     

Fibonacci and Lucas congruences and their applications

In this paper we obtain some new identities containing Fibonacci and Lucas numbers. These identities allow us to give some congruences concerning Fibonacci and Lucas numbers such as L _2mn+k ≡ (−1)^(m+1)n L _k (mod L _m ), F _2mn+k ≡ (−1)^(m+1)n F _k (mod L _m ), L _2mn+k ≡ (−1) ^mn L _k (mod F _m ) and F _2mn+k ≡ (−1) ^mn F _k (mod F _m ). By the achieved identities, divisibility properties of Fibonacci and Lucas numbers are given. Then it is proved that there is no Lucas number L _n such that L _n = L ₂ ^k t L _m x ² for m > 1 and k ≥ 1. Moreover it is proved that L _n = L _m L _r is impossible if m and r are positive integers greater than 1. Also, a conjecture concerning with the subject is given.     

Mean Ramsey–Turán numbers

A ρ‐mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most ρ. For a graph H and for ρ ≥ 1, the mean Ramsey–Turán number RT(n, H,ρ ? mean) is the maximum number of edges a ρ‐mean colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. It is conjectured that where is the maximum number of edges a k edge‐colored graph with n vertices can have under the condition it does not have a monochromatic copy of H. We prove the conjecture holds for . We also prove that . This result is tight for graphs H whose clique number equals their chromatic number. In particular, we get that if H is a 3‐chromatic graph having a triangle then . © 2006 Wiley Periodicals, Inc. J Graph Theory 53: 126–134, 2006     

The Euler and Springer numbers as moment sequences

I study the sequences of Euler and Springer numbers from the point of view of the classical moment problem.     

The odd-number sequence: squares and sums

Direct study of various characteristics of integers and their interactions is readily accessible to undergraduate students. Integers obviously fall in different classes of modular rings and thus have features unique to that class which can result in a variety of formations, particularly with sums of squares. The sum of the first n odd numbers is itself the square of n within the odd number sequence, from which testing for primality within the Fibonacci sequence is investigated in this note.