{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "\"drawing\"\n", "

Experimental Mathematics Using SageMath — AIMS-ZA-2024-25

\n", "\n", "\n", "## Instructors: \n", "\n", "* **Evans Ocansey**" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Day 11 — Positive Divisors\n", "\n", "[comment]: <> (

Day 02 — Introduction to SageMath: A Mathematics Software for All

)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "The outline of the this notebook is as follows:\n", "\n", "## Table of Contents: \n", "* [ ] [Investigating Arithmetic Functions](#recall-from-last-lecture)\n", "* [ ] [Arithmetic Functions](#the-partition-class-using-keywords)\n", " * [Multiplicative Functions](#keyword-max-slope)\n", " * [Completely Multiplicative Functions](#keyword-parts-in)\n", "* [ ] [Multiplicative Properties of $\\tau$ and $\\sigma$](#statements-to-explore)\n", "* [ ] [Exploring $\\tau$ and $\\sigma$ in Sage](#statements-to-explore)\n", "* [ ] [Questions to Explore](#the-partition-class-using-keywords)\n", " * [Exploring Question 1](#exploring-question-1)\n", " * [Exploring Question 2](#exploring-question-2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "# Investigating Arithmetic Functions \n", "\n", "Today, we will explore the functions $\\tau$ and $\\sigma$, where:\n", "- $\\tau(n)$ counts the number of positive divisors of $n$, and\n", "- $\\sigma(n)$ gives the sum of the positive divisors of $n$.\n", "\n", "Our goal is to use logical reasoning, supported by **SageMath**, to not only make conjectures but also begin proving them." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Arithmetic Functions \n", " \n", "**Definition**: An **arithmetic function** is a real or complex-valued function $f$ defined on the set of positive integers:\n", "\n", "$$\n", "f : \\mathbb{Z}_{>0} \\to \\mathbb{C}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Multiplicative Functions \n", "\n", "An arithmetic function $ f $ is said to be **multiplicative** if it satisfies the following conditions:\n", "1. $ f $ is not identically zero, and\n", "2. $f(a \\cdot b) = f(a) \\cdot f(b) \\quad \\text{whenever } \\gcd(a, b) = 1. $" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Completely Multiplicative Functions \n", "\n", "An arithmetic function $ f $ is **completely multiplicative** if:\n", "\n", "$$\n", "f(a \\cdot b) = f(a) \\cdot f(b) \\quad \\text{for all } a, b \\in \\mathbb{N}.\n", "$$" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Multiplicative Properties of $\\tau$ and $\\sigma$\n", "\n", "Are $\\tau$ and $\\sigma$ multiplicative?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Case of $\\tau$ \n", "\n", "Let us prove that $\\tau$ is a multiplicative function. To do so, we will consider the following questions:\n", "\n", "1. **Is $\\tau$ an arithmetic function?**\n", " - Yes, $\\tau$ is an arithmetic function. For all $ n \\in \\mathbb{N} $, $ \\tau(n) $ is a natural number. Since $ \\mathbb{N} \\subset \\mathbb{C} $, it follows that $\\tau$ is an arithmetic function.\n", "\n", "2. **For $ a, b \\in \\mathbb{N} $ with $ \\gcd(a, b) = 1 $, is $ \\tau(a \\cdot b) = \\tau(a) \\cdot \\tau(b) $?**\n", " - To prove this, consider:\n", " - The number of divisors of $ a \\cdot b $, where $ \\gcd(a, b) = 1 $.\n", " - Since $ \\gcd(a, b) = 1 $, the divisors of $ a \\cdot b $ are exactly the combinations of the divisors of $ a $ and $ b $. Specifically:\n", " - For a divisor $ a_1 $ of $ a $, the divisors of $ a \\cdot b $ include $ a_1 $ multiplied by each divisor of $ b $.\n", " - The total number of such divisors is $ \\tau(a) \\cdot \\tau(b) $, proving that $ \\tau(a \\cdot b) = \\tau(a) \\cdot \\tau(b) $.\n", "\n", "Thus, $\\tau$ is multiplicative." ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### The Case of $\\sigma$ \n", "\n", "Write your proof here." ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Exploring $\\tau$ and $\\sigma$ in Sage \n", "\n", "How do you define $\\tau$ and $\\sigma$ in Sage?" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Questions to Explore " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exploring Question 1 " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exploring Question 2 \n" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Conjectures to Explore " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exploring Conjecture 1 " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Exploring Conjecture 2 " ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 10.4", "language": "sage", "name": "sagemath" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.11.2" } }, "nbformat": 4, "nbformat_minor": 4 }