The first dynamical system; Random Number Theory – Ramsey Doccourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Title: The first dynamical system; and Random Number Theory

Lecturer: Carl Pomerance

Date: November 8, 2016

Main Topics: Chains with $\sigma$, distribution of primes, randomness in math

Definitions: Amicable, Perfect, Abundant, Deficient


Introduction

There were two talks given on November 8, 2016. The first (“the first dynamical system”) was about the natural numbers and the function which sums its divisors. The second (“Random number theory”) discusses the value of using randomness in number theory and mathematics.

The slides for both talks are included as links. The second talk was recorded and will be linked to as soon as it is published.

My notes are sparse because there were slides and the second talk was recorded. Instead of including detailed notes, I’ve included some extra problems about these topics.

The first dynamical system

Here are the slides from the talk [PDF].

Carl Pomerance has many other talks on his website.

The talk primarily concerns the function $\sigma(n)$ which sums the proper divisors of a natural number $n$. For example,

  • $\sigma(6)= 1 + 2 + 3 = 6$ (A perfect number)
  • $\sigma(10)= 1 + 2 + 5 = 8 < 10$ (A deficient number)
  • $\sigma(12)= 1 + 2 + 3 + 4 + 6 = 16 > 12$ (An abundant number)
Definition A natural number $n > 1$ is defined to be

  • perfect if $\sigma(n) = n$,
  • abundant if $\sigma(n) > n$,
  • deficient if $\sigma(n) < n$,

A pair of natural numbers $n,m$ are amicable if $\sigma(n) = m$ and $\sigma(m)=n$.

Project Euler problems

Project Euler (an online collection of math related programming problems) has many problems related to $\sigma$, abundant numbers and amicable pairs. Here are some of them to give you a feel for these objects.

  1. Problem 21. Amicable numbers.
  2. Problem 23. Non-abundant sums.
  3. Problem 95. Amicable chains.
  4. Problem 69 and Problem 70. These are about the related function $\phi$.

Random Number Theory

Here are the slides [PDF].

Carl Pomerance has many other talks on his website.

The Paul Erdős misquote

Carl Pomerance described the origin of the quote misattributed to Paul Erdős:

Einstein: “God does not play dice with the universe.”
Paul Erdős-Kac: Maybe so, but something is going on with the primes.

The intention was that the Paul Erdős-Kac theorem says something about the distribution of the primes, not that Paul Erdős and Kac themselves has said this (Note the lack of quotation marks!). Wikiquotes has a good description of the story.

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