Amicable numbers are two different numbers where each number is the sum of the proper divisors of the other. Proper divisors are the positive divisors of a number, excluding the number itself. For example, the numbers 220 and 284 are amicable because the proper divisors of 220 add up to 284, and the proper divisors of 284 add up to 220. This concept has been studied since ancient times and is linked to number theory. Mathematicians like Pythagoras and Euclid explored these relationships, and amicable numbers are often used to illustrate the beauty and complexity of numbers in mathematics.