$ϕ$ 和 $σ$ 迭代的原像数量

The investigation of Euler's totient function preimages

We propose a lower estimation for computing quantity of the inverses of Euler's function. We answer the question about the multiplicity of $m$ in the equation $φ(x) = m$ \cite{Ford}. An analytic expression for exact multiplicity of $m = {{2}^{{{2}^{n}}+a}}$, where $a\in N$, $a<{{2}^{n}}$, $φ(t)={{2}^{{{2}^{n}}+a}}$ was obtained. A lower bound of inverses number for arbitrary $m$ was found. New numerical metric was proposed.

Some remarks on Euler's totient function

The image of Euler's totient function is composed of the number 1 and even numbers. However, many even numbers are not in the image. We consider the problem of finding those even numbers which are in the image and those which are not. If an even number is in the image, then its preimage can have at most half its elements odd. However, it may contain only even numbers. We consider the structure of the preimage of certain numbers in the image of the totient function.

$(\mathbb{P}^1)^n$ 上满射自同态的原像问题

Preimages Question for Surjective Endomorphisms on $(\mathbb{P}^1)^n$

本文证明在数域上 $(\mathbb{P}^1)^n$ 的支配自同态中，不变子簇的原像差集最终无 $K$-有理点，从而解决该情形下的原像问题。

Transcendence of values of the iterated exponential function at algebraic points

We say that the order of an algebraic number $A$ is the minimum of positive integers $k$ such that $A^k$ is rational. In this paper, we show that the number of algebraic numbers $A$ with order $k$ such that \[ A,\ A^A,\ A^{A^A},\ \ldots \] converges to an algebraic number is approximated by $(e-1/e)φ(k)$. Here $φ(k)$ denotes Euler's totient function.

自然数上的一族迭代映射

A Family of Iterated Maps on Natural Numbers

本文研究自然数上由素数与合数分支规则定义的一族迭代映射，分析其轨道的周期行为，并给出部分证明及若干基于数值证据的开放问题。

PCF多项式树形表示的超群

Overgroups of the arboreal representation of PCF polynomial

本文研究了PCF多项式树形Galois表示的像所在的自然超群，发现特定新PCF多项式的表示像同构于该超群，并证明其正规子群构成唯一主列以界定生成元数量。

Divisor-sum fibers

Let $s(\cdot)$ denote the sum-of-proper-divisors function, that is, $s(n) = \sum_{d\mid n,~d<n}d$. Erdős-Granville-Pomerance-Spiro conjectured that for any set $\mathcal{A}$ of asymptotic density zero, the preimage set $s^{-1}(\mathcal{A})$ also has density zero. We prove a weak form of this conjecture: If $ε(x)$ is any function tending to $0$ as $x\to\infty$, and $\mathcal{A}$ is a set of integers of cardinality at most $x^{\frac12+ε(x)}$, then the number of integers $n\le x$ with $s(n) \in \mathcal{A}$ is $o(x)$, as $x\to\infty$. In particular, the EGPS conjecture holds for infinite sets wit…

The number of iterates of the Carmichael lambda function required to reach 1

The Carmichael lambda function $λ(n)$ is defined to be the smallest positive integer $m$ such that $a^m \equiv 1 \pmod{n}$ for all $(a,n)=1.$ $λ_k(n)$ is defined to be the $k$th iterate of $λ(n).$ Let L(n) be the smallest $k$ for which $λ_k(n)=1.$ It's easy to show that $L(n) \ll \log n.$ It's conjectured that $L(n)\asymp \log\log n,$ but previously it was not known to be $o(\log n)$ for almost all $n.$ We will show that $L(n) \ll (\log n)^δ$ for almost all $n,$ for some $δ<1.$ We will also show $L(n) \gg \log\log n$ for almost all $n$ and conjecture a normal order for L(n).

The iterated Carmichael λ-function and the number of cycles of the power generator

Iteration of the modular l-th power function f(x) = x^l (mod n) provides a common pseudorandom number generator (known as the Blum-Blum-Shub generator when l=2). The period of this pseudorandom number generator is closely related to λ(λ(n)), where λ(n) denotes Carmichael's function, namely the maximal multiplicative order of any integer modulo n. In this paper, we show that for almost all n, the size of λ(λ(n)) is n/exp((1+o(1))(log log n)^2 log log log n). We conjecture an analogous formula for the k-th iterate of λ. We deduce that for almost all n, the psuedorandom number generator described…

A problem of Ramanujan, Erdos and Katai on the iterated divisor function

We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.

Iterates of Generic Polynomials and Generic Rational Functions

In 1985, Odoni showed that in characteristic $0$ the Galois group of the $n$-th iterate of the generic polynomial with degree $d$ is as large as possible. That is, he showed that this Galois group is the $n$-th wreath power of the symmetric group $S_d$. We generalize this result to positive characteristic, as well as to the generic rational function. These results can be applied to prove certain density results in number theory, two of which are presented here. This work was partially completed by the late R.W.K. Odoni in an unpublished paper.

Using Iterated Function Systems to Reveal Biases in the Distribution of Prime Numbers

Iterated function systems (IFS) can be a surprisingly useful tool for studying structure in data. Here we present results stemming from a 2013 computational study by the author using IFS. The results include fractal patterns that reveal "repulsive" phenomena among primes in a wide range of classes, having specified arithmetic or congruence properties. Some of the phenomena shown in our computations relate to the recent, groundbreaking work of Lemke Oliver and Soundararajan on biases between consecutive primes. We do not have asymptotics to explain our results, but provide graphs, data, and det…

Iterated Chvatal-Gomory Cuts and the Geometry of Numbers

Chvatal-Gomory cutting planes (CG-cuts for short) are a fundamental tool in Integer Programming. Given any single CG-cut, one can derive an entire family of CG-cuts, by `iterating' its multiplier vector modulo one. This leads naturally to two questions: first, which iterates correspond to the strongest cuts, and, second, can we find such strong cuts efficiently? We answer the first question empirically, by showing that one specific approach for selecting the iterate tends to perform much better than several others. The approach essentially consists in solving a nonlinear optimization problem o…

Overlapping iterated function systems from the perspective of Metric Number Theory

In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation. This result shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems our results apply to include those arising from Bernoulli convol…