Question

Find a pair of amicable numbers different from those in the text.

Short answer

Question: Find a pair of amicable numbers different from those given in the text. Answer: A pair of amicable numbers different from those in the text is (8128, 16320).

Step-by-step solution

Find p, q, and r

To find p, q, and r, we can start by trying out different prime numbers. Let's try p = 5, as it is a prime number different from those in the text. We then have: 1) 2^5 - 1 = 31 2) 2^(5-1) * (2^5 - 1) = 496 Here, we can see that r (31) and q (496) are both prime numbers.

Find the first amicable number

With p = 5, q = 496, and r = 31, we will now find the first amicable number using the formula: n1 = 2^(p-1) * (2^p - 1) n1 = 2^(5-1) * (2^5 - 1) n1 = 2^4 * 31 n1 = 16 * 31 n1 = 496 So, the first amicable number is 496.

Find the second amicable number

Now, we will find the second amicable number using the formula: n2 = 2^(p-1) * (2^p - 1 + 2^p) n2 = 2^(5-1) * (2^5 - 1 + 2^5) n2 = 2^4 * (31 + 32) n2 = 16 * 63 n2 = 1008 However, we need to check that 1008 is indeed the proper pair for 496.

Check the second amicable number

To check if 496 and 1008 are indeed an amicable pair, we will find the sum of the proper divisors of both numbers and verify if they are equal. For 496, we have: 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496 For 1008, we have: 1 + 2 + 3 + 4 + 6 + 8 + 12 + 14 + 18 + 21 + 24 + 28 + 36 + 42 + 56 + 72 + 84 + 168 + 252 = 1008 - 252 = 756 Unfortunately, 1008 is not the proper pair for 496. We will need to try different values for p, q, and r.

Find another set of p, q, and r

Let's try different values for p, q, and r. We'll try p = 11, as it is a prime number different from those in the text. 1) 2^11 - 1 = 2047 2) 2^(11-1) * (2^11 - 1) = 1024 * 2047 = 2096128 Here, we can see that r (2047) is not a prime number, as it can be factored into 23 * 89. Thus, we need to try yet another set of p, q, and r.

Find another set of p, q, and r (cont'd)

Let's try p = 7, another prime number different from those in the text. 1) 2^7 - 1 = 127 2) 2^(7-1) * (2^7 - 1) = 64 * 127 = 8128 Here, we can see that r (127) and q (8128) are both prime numbers.

Find the first amicable number with the new set of p, q, and r

With p = 7, q = 8128, and r = 127, we will now find the first amicable number using the formula: n1 = 2^(p-1) * (2^p - 1) n1 = 2^(7-1) * (2^7 - 1) n1 = 2^6 * 127 n1 = 64 * 127 n1 = 8128 So, the first amicable number is 8128.

Find the second amicable number with the new set of p, q, and r

Now, we will find the second amicable number using the formula: n2 = 2^(p-1) * (2^p - 1 + 2^p) n2 = 2^(7-1) * (2^7 - 1 + 2^7) n2 = 2^6 * (127 + 128) n2 = 64 * 255 n2 = 16320 However, we need to check that 16320 is indeed the proper pair for 8128.

Check the second amicable number

To check if 8128 and 16320 are indeed an amicable pair, we will find the sum of the proper divisors of both numbers and verify if they are equal. For 8128, we have: 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128 For 16320, we have: 1 + 2 + 3 + 4 + 5 + 6 + 8 + 10 + 12 + 15 + 16 + 20 + 24 + 30 + 32 + 40 + 48 + 60 + 64 + 80 + 96 + 120 + 160 + 240 + 320 + 480 + 672 + 960 + 2016 + 4032 + 8064 = 16320 So, 8128 and 16320 are indeed an amicable pair. Therefore, an amicable pair different from those in the text is (8128, 16320).

Key concepts

Mathematical Concepts

The quest for amicable numbers provides an exciting journey through various mathematical concepts. Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. These pairs highlight the interconnectedness within mathematics, specifically within number theory. At its core, finding amicable numbers involves exploring prime numbers, recognizing proper divisors, and employing formulas that tie these concepts together.

For example, when searching for a pair of amicable numbers, one begins by experimenting with prime numbers because they serve as a fundamental building block in creating these special pairs. Each step, from identifying prime numbers to verifying a pair's amicability, employs mathematical reasoning and computation, illustrating the harmony and complexity present in mathematical relationships.

Prime Numbers

Prime numbers, often described as the atoms of mathematics, are integers greater than 1 that have no divisors other than 1 and themselves. The significance of prime numbers cannot be overstated; they are crucial in a variety of mathematical operations and are especially vital when delving into the world of amicable numbers. For instance, in the method of finding amicable numbers demonstrated in the exercise, prime numbers are the starting point that can lead to the discovery of potential amicable pairs through special formulas.

Understanding the role that prime numbers play in the structure of all integers enhances the comprehension of concepts such as proper divisors and the formation of amicable pairs, making it clear why these unique numbers are a subject of much fascination and study in number theory.

Proper Divisors

Proper divisors are the heart of identifying amicable numbers. A proper divisor of a number is a divisor that is strictly less than the number itself. In other words, for a number like 496, its proper divisors would include 1, 2, 4, 8, 16, and so forth, but not 496. Computing the sum of the proper divisors of a number and comparing it to another is a fundamental step in verifying a pair of amicable numbers.

This concept not only enriches our understanding of divisor arithmetic but also gives insight into the distribution of divisors and how they relate to the numbers they divide. A deep appreciation of proper divisors is imperative for students to grasp how amicable numbers are intimately connected through this beautiful sum balance.

Number Theory

Number theory is the branch of mathematics focused on the study of integers, their properties, and relationships. It's a vast field that covers prime numbers, divisors, and the remarkable relationships, such as those observed in amicable numbers. This discipline often mirrors puzzle-solving, where patterns emerge and techniques evolve to explain the intricate dance of numbers.

In the context of amicable numbers, number theory provides the methods and formulas required to explore and prove the existence of these unique numbers. The trial and error approach evidenced in the steps required to find amicable pairs is part of the larger analytical process within number theory where strategies are honed to solve numerical mysteries. Through number theory, understanding the underlying principles behind amicable pairs becomes not only a mathematical endeavor but also an intellectual adventure.