Problem 23 - A History of Mathematics: An Introduction

Question

Show that 1184 and 1210 are amicable numbers that are not a consequence of the theorem of Th?bit ibn Qurra.

Step-by-Step Solution

Answer

Question: Show that 1184 and 1210 are amicable numbers, and that they are not a consequence of the theorem of Thabit ibn Qurra. Answer: 1184 and 1210 are amicable numbers because the sum of the proper divisors of 1184 equals 1210, and the sum of the proper divisors of 1210 equals 1184. However, they do not conform to Thabit ibn Qurra's theorem, as we couldn't find any appropriate value of n where 1184 or 1210 conform to the requirements of the theorem.

Step 1: Find the sum of divisors for each number

Let us find the proper divisors of 1184 and 1210 and calculate the sum of these divisors. Divisors of 1184 (other than the number itself): 1, 2, 4, 8, 16, 74, 148, 296, 592 Sum of divisors of 1184: 1 + 2 + 4 + 8 + 16 + 74 + 148 + 296 + 592 = 1210 Divisors of 1210 (other than the number itself): 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605 Sum of divisors of 1210: 1 + 2 + 5 + 10 + 11 + 22 + 55 + 110 + 121 + 242 + 605 = 1184 Since the sum of the proper divisors of 1184 equals 1210, and the sum of the proper divisors of 1210 equals 1184, 1184 and 1210 are amicable numbers.

Step 2: Check for conformity with Thabit ibn Qurra's theorem

Assume the theorem holds for these amicable numbers, i.e., there exists an n such that either 1184 or 1210 can be expressed as \(\frac{1}{2}ab(a^2+b^2+c^2 -ab-ac-bc)\), with a, b, and c satisfying the theorem conditions mentioned in the analysis. Firstly, let's check if any combination of a, b, and c (where \(a = 3 \cdot 2^{n-1} - 1\), \(b = 3 \cdot 2^{n-1}\), and \(c = 3 \cdot 2^{n-1} + 1\)) can generate an odd product \(abc\). Since b is always even (since it is a multiple of 3 and a power of 2), the product \(abc\) will always be even. The numbers 1210 and 1184 are both even, so this requirement is satisfied. Now, we need to see if we can find an n where the amicable numbers conform to the theorem. We know that the smallest amicable number in a pair will be \(\frac{1}{2}ab(a^2+b^2+c^2 -ab-ac-bc)\), which means that its value will always be an integer multiple of \(ab\), because \(abc\) divides \((a^2+b^2+c^2 -ab-ac-bc)\). However, the prime factorization of 1184 is 2^4 * 74, which contains no integer multiples of the form \(ab\), where \(a = 3 \cdot 2^{n-1} - 1\) and \(b = 3 \cdot 2^{n-1}\). We can confirm this by testing for values of n, where n ∈ {1,2,3,4} (testing larger values would lead to amicable pairs exceeding 1184). In none of these cases, we find an appropriate multiple. Since we couldn't find any appropriate value of n where 1184 or 1210 conform to the requirements of Thabit ibn Qurra's theorem, we conclude that the amicable pair 1184 and 1210 is not a consequence of the theorem.

Key Concepts

Thabit ibn Qurra's Theorem

Thabit ibn Qurra's Theorem provides a method to generate amicable numbers, which are pairs of numbers where each number is the sum of the proper divisors of the other. The pair of numbers, when checked against the theorem, can sometimes confirm that they conform to a specific pattern of generation. Therefore, understanding this theorem is essential for identifying patterns in such numerical sequences.

Thabit ibn Qurra's method involves selecting values for three variables, denoted as \(a\), \(b\), and \(c\), based on the equation:

• \(a = 3 \cdot 2^{n-1} - 1\)
• \(b = 3 \cdot 2^{n-1}\)
• \(c = 3 \cdot 2^{n-1} + 1\)

With these values, the amicable numbers are tested against another formula, \[\frac{1}{2}ab(a^2+b^2+c^2 -ab-ac-bc)\]where the numbers are generated as multiples of certain expressions.

However, not all amicable numbers fit the structure dictated by Thabit's formula. As seen with the numbers 1184 and 1210, which do not arise from this theorem, demonstrating its fascinating complexity. It's important to explore the theorem while keeping in mind that exceptions do exist, as demonstrated in this exercise.

Divisors

The concept of divisors is fundamental in understanding amicable numbers. A divisor of a number is any integer that divides the number without leaving a remainder. Divisors can be proper, which means they exclude the number itself, and they play an essential role in identifying amicable numbers.

For instance, when you consider the numbers 1184 and 1210:

• The proper divisors of 1184 are: 1, 2, 4, 8, 16, 74, 148, 296, 592
• The proper divisors of 1210 are: 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605

By calculating the sum of their proper divisors, it turns out that:

• The sum of the divisors of 1184 is 1210
• The sum of the divisors of 1210 is 1184

This mutual property confirms their status as amicable numbers.

Recognizing and summing divisors allows mathematicians to uncover these unique relationships among numbers. Therefore, understanding divisors not only extends to tackling exercises involving amicable numbers but also builds a strong foundation in number theory.

Number Theory

Number theory is a branch of mathematics dedicated to understanding the properties and relationships of numbers, particularly integers. It includes studying concepts such as divisors, prime numbers, and their complex interactions. Amicable numbers are an intriguing focus within number theory, because they highlight an unexpected and harmonious relationship between seemingly unrelated integers.

Number theory has deep historical roots, dating back to ancient mathematicians, but it remains relevant as it finds applications in areas like cryptography and algorithm design. Often, number theory addresses highly abstract and theoretical questions that, over time, find practical use. For students, engaging with number theory through amicable numbers offers a glimpse into the broader and often unexpected beauty of mathematics.

Moreover, theories like Thabit ibn Qurra's Theorem emerge from this field, providing methods to generate pairs of amicable numbers. Such inquiries demonstrate how timeless mathematical thoughts persist in modern studies. Therefore, number theory not only fosters a sense of numerical curiosity but also develops a deeper appreciation for the interconnectedness of math concepts.