Generalizing Abundancy Index to Gaussian Integers

10的每个朋友至少有10个不同的素因子

Each friend of 10 has at least 10 nonidentical prime factors

本文应用Nielsen 2007年的方法证明10的每个朋友至少有10个不同的素因子，改进了对10的朋友的素因子个数下界。

关于“广义丰度指数”的若干观察

Some Observations about the "Generalized Abundancy Index"

本文研究由交换置换元组定义的广义丰度指数，利用概率模型证明其均值极限为 $ζ(2)\cdots ζ(\ell)$，并对 $\ell=2$ 情形的渐近偏差提出精确猜想。

Multiperfect Numbers in Certain Quadratic Rings

Using an extension of the abundancy index to imaginary quadratic rings that are unique factorization domains, we investigate what we call $n$-powerfully $t$-perfect numbers in these rings. This definition serves to extend the concept of multiperfect numbers that have been defined and studied in the integers. At the end of the paper, as well as at various points throughout the paper, we point to some potential areas for further research.

On the Radical of Multiperfect Numbers and Applications

It is conjectured that for a perfect number $m,$ $\rm{rad}(m)\ll m^{\frac{1}{2}}.$ We prove bounds on the radical of multiperfect number $m$ depending on its abundancy index. Assuming the ABC conjecture, we apply this result to study gaps between multiperfect numbers, multiperfect numbers represented by polynomials. Finally, we prove that there are only finitely many multiperfect multirepdigit numbers in any base $g$ where the number of digits in the repdigit is a power of $2.$ This generalizes previous works of several authors including O. Klurman, F. Luca, P. Polack, C. Pomerance and others.

On 2-powerfully Perfect Numbers in Three Quadratic Rings

Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call $n$-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of $2$-powerfully perfect numbers in the rings $\mathcal O_{\mathbb{Q}(\sqrt{-1})}$, $\mathcal O_{\mathbb{Q}(\sqrt{-2})}$, and $\mathcal O_{\mathbb{Q}(\sqrt{-7})}$, the three imaginary quadratic rings with unique factorization in which $2$ is not a prime.

Index divisibility in the orbit of 0 for integral polynomials

Let $f(x) \in \bbz[x]$ and consider the index divisibility set $D = \{n \in \bbn : n \mid f^n(0)\}$. We present a number of properties of $D$ in the case that $(f^n(0))_{n=1}^\infty$ is a rigid divisibility sequence, generalizing a number of results of Chen, Stange, and the first author. We then study the polynomial $x^d + x^e + c \in \bbz[x]$, where $d > e \ge 2$ and determine all cases where this map has a finite index divisibility set.

Symplectic modular symbols

Let K be a number field with euclidean ring of integers O. Let G be a finite-index torsion-free subgroup of Sp(2n, O). We exhibit a finite, geometrically defined spanning set of the top dimensional integral cohomology of G by generalizing the modular symbol algorithm of Ash and Rudolph.

Solving the Odd Perfect Number Problem: Some Old and New Approaches

A perfect number is a positive integer $N$ such that the sum of all the positive divisors of $N$ equals $2N$, denoted by $σ(N) = 2N$. The question of the existence of odd perfect numbers (OPNs) is one of the longest unsolved problems of number theory. This thesis presents some of the old as well as new approaches to solving the OPN Problem. In particular, a conjecture predicting an injective and surjective mapping $X = σ(p^k)/p^k, Y = σ(m^2)/m^2$ between OPNs $N = {p^k}{m^2}$ (with Euler factor $p^k$) and rational points on the hyperbolic arc $XY = 2$ with $1 < X < 1.25 < 1.6 < Y < 2$ and $2.8…

d'Arcais 数的大偏差理论

Large Deviations for the d'Arcais Numbers

研究d'Arcais多项式系数的大偏差行为，推导出其大偏差率函数并关联到过剩度指数信息。