Problem 22 - A History of Mathematics: An Introduction

Question

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

Step-by-Step Solution

Answer

Answer: Yes, the numbers 17,296 and 18,416 are amicable according to Ibn Qurra's theorem.

Step 1: Identify the prime factors of 17,296 and 18,416

First, let's find the prime factors of the given numbers: 17,296 = 2^5 x 541 18,416 = 2^4 x 1,151 Now that we have identified the prime factors of the numbers, let's see if these numbers can be written in the form as stated in the theorem for any prime numbers p and q.

Step 2: Check if we can find primes p and q according to the theorem

The theorem states that we should be able to write the numbers as m=n(p-1) and n=m=q-1=q for some prime numbers p and q. From step 1, we can see the relationship of the prime factors: 17,296 = 18,416 x (2^1 - 1) 18,416 = 17,296 x (1,151 - 1) Here, the primes p and q are 2 and 1,151, respectively. Since we can write the numbers in the required way using these primes, the numbers satisfy Ibn Qurra's theorem.

Step 3: Conclude that the numbers are amicable

As we have found prime numbers p and q and written the given numbers according to Ibn Qurra's theorem, we can conclude that the numbers are amicable (friendly) because they satisfy the condition stated in the theorem. Therefore, 17,296 and 18,416 are amicable numbers.

Key Concepts

Prime Factorization

Prime factorization is the process of finding the set of prime numbers which, when multiplied together, result in the original number. This is an essential concept in mathematics since any integer greater than 1 can be factored uniquely into prime numbers. For example, when we factorize the number 17,296, we express it as a product of prime numbers:

• 17,296 = 2^5 \( \times \) 541

Here, 2^5 and 541 are the prime factors of 17,296.
Similarly, for 18,416, its prime factorization is:

• 18,416 = 2^4 \( \times \) 1,151

These factor sets are important because they reveal structural properties of the number, such as divisibility and the relationship to other numbers.
In the case of amicable numbers, prime factorization is a crucial step in verifying those numbers as special or amicable pairs, according to Ibn Qurra’s theorem.

Ibn Qurra's Theorem

Ibn Qurra's Theorem is a method used to identify pairs of amicable numbers. These are two different numbers where each number is the sum of the proper divisors (excluding the number itself) of the other. The theorem provides specific conditions and formulas involving prime numbers under which numbers can be considered amicable.
Ibn Qurra indicated that, for two numbers to be amicable, they must relate through a specific formula involving primes, such that:

• If 17,296 can be expressed as \( n(p-1) \) and
• 18,416 = \( m(q-1) \)

Where \( p \) and \( q \) are primes.
In the example, primes were used to show that 17,296 = 18,416 * (2^1 - 1) and 18,416 = 17,296 * (1,151 - 1), demonstrating the relationship demanded by the theorem. These relationships confirm that 17,296 and 18,416 form an amicable pair, proving how Ibn Qurra’s theorem is applied in practice.

Number Theory

Number theory is a branch of pure mathematics devoted to the study of the properties and relationships of numbers, particularly integers. Topics in number theory often involve problems related to finding patterns, solving equations, and understanding sequences of numbers.
Amicable numbers are an interesting concept within number theory and date back to ancient mathematics, showcasing the relationships between numbers in unique ways.
The study of such numbers can reveal surprising and deep links between different number concepts such as divisors and prime numbers. Many mathematical theorems, such as Ibn Qurra’s, have been developed from these concepts, allowing mathematicians to classify numbers more effectively and explore their intricate connections.
Understanding the application of number theory is not only pivotal in academic circles but also applies to modern-day computing and cryptography.