Math Musings

  Two distinct positive integers a and b are said to form an amicable pair if the sum of the proper divisors of a is equal to b , and the sum of the proper divisors of b is equal to a . A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1,2, and 3. The ancient Greeks were already interested in amicable pairs, and they knew of an example: 220 and 284. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. The proper divisors of 284 are 1, 2, 4, 71, and 142. If we write s(n) for the sum of the proper divisors of n. We have 220 and 284 was the only amicable pair known to the ancient Greeks, but over the centuries, two more pairs were found: 17296 and 18416, and 9363584 and 9437056. Those three pairs were all that were known before Euler tackled the problem of finding amicable pairs, and he found 58 more pairs, thus increasing the supply of known pairs from 3 all the way up to 61.