Show that 17,296 and 18,416 are amicable by using ibn Qurra's theorem.

Name: Problem 22, Chapter 9: A History of Mathematics: An Introduction
Uploaded: 2020-06-13T20:47:14-08:00
Duration: 4 min 42 s
Channel: Kira Harland
Description: Show that 17,296 and 18,416 are amicable by using ibn Qurra's theorem.

Show that 17,296 and 18,416 are amicable by using ibn...

Key Concepts

Amicable Numbers

Amicable numbers are two different positive integers such that the sum of the proper divisors of each is equal to the other number. This concept is central in number theory and reflects a deep and classical curiosity about numbers that are in a kind of harmonious relationship with each other through their divisor structure.

Ibn Qurra's Theorem

Ibn Qurra's theorem is an early method for generating amicable pairs. It provides a formula or a set of conditions involving prime numbers that, when satisfied, ensure that the two numbers constructed from these primes are amicable. This theorem not only exemplifies an ingenious approach to detecting such relationships in numbers but also highlights historical advancements in number theory.

Divisor Sum Function

The divisor sum function, often denoted by ?(n) when including n itself or s(n) when considering proper divisors, is fundamental in studying amicable numbers. It maps a number to the sum of its divisors and is pivotal in checking whether two numbers are amicable—if the sum of the proper divisors of one equals the other, and vice versa, then the numbers form an amicable pair.

Transcript

00:01 Hi everybody.

00:02 Today we're going to be trying to prove the alternate exterior angle converse theorem.

00:07 What this is saying is that if we know angle one is congruent to angle two, if we are given that information, how can we say that m is now parallel to m? and so you would be wondering, where is m and n? m, we will call the top one here.

00:36 And n, let's call the bottom line here.

00:38 And this line here is the transversal, of course.

00:41 Whenever we do a proof, we want to make a statement table.

00:45 So we are going to create a statement table.

00:55 And as we create the statement table, i want you to think about what are some of the givens you know, what are some of the things just by looking at the diagram now that you know? okay.

01:06 And we will put statements over here.

01:14 And with every statement we make, we always need to conclude our reasoning.

01:19 So our reasons will go here.

01:20 The first thing we want to say is just state our givens.

01:24 So we are given that angle one is congruent to angle two.

01:31 And this is just given.

01:36 Now there's something else just by looking at this diagram that you might already be thinking.

01:41 And that is that angle one is congruent to angle three.

01:47 And this is because vertical angles are congruent.

02:00 Now, without even looking at the diagram, what can you now say about angles 2 and 3? if you said that they are congruent, you are right, and that's by the transitive property.

02:14 So i know that angles 2 and angles 3 are congruent because of the transitive property...

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