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What is extremely special about these two numbers: 220 and 284? They are ______ pairs.
The numbers 220 and 284 are known as amicable numbers, meaning that they are a pair of numbers where each number is the sum of the proper divisors of the other number. In this case, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which add up to 284. Similarly, the proper divisors of 284 are 1, 2, 4, 71, and 142, which add up to 220.
The discovery of amicable numbers dates back to ancient Greece, but the specific pair of 220 and 284 was first recorded by the Pythagorean philosopher Nicomachus in the 2nd century AD. They were later studied by other mathematicians, including the famous mathematician Leonhard Euler in the 18th century.
Yes, there are many other pairs of amicable numbers. In fact, it is estimated that there are over 1 million pairs of amicable numbers, although only a few hundred have been discovered so far. The smallest known pair is 220 and 284, and the largest known pair is 9,363,584,538,952,064, and 9,437,265,624,385,600.
The main reason these numbers are considered special is because of their unique relationship as amicable numbers. They also have some interesting mathematical properties, such as being part of a set of numbers called perfect numbers, which are numbers that are equal to the sum of their proper divisors.
Amicable numbers have been studied and used in various areas of mathematics, including number theory, algebra, and geometry. They also have applications in cryptography and coding theory. In addition, the study of amicable numbers has led to the discovery of other types of special numbers, such as sociable numbers and perfect numbers.