Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself.For example, the smallest pair of amicable numbers is (220, 284); for the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71 and 142, of which the sum is 220.
The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564)...
Ancient philosophers/mathematicians used friendly numbers to describe friendships between people (Aristotle in his work Ethics, for example). A pair of friendly numbers became a symbol of friendship.
Although the equation that derives pairs of amicable numbers was popularized by Fermat and Descartes in the early 16th century, the concept and derivation of these unique numbers was first investigated by an Iranian mathematician named Thābit ibn Qurra. Qurra was also one of the first mathematicians to describe the phenomenon that we now know as the Pythagorean Theorem.
Thabit ibn Qurra Ibrahim, who lived in Baghdad in the 9th century, came across an algorithm by which he was able to find more amicable pairs. He did operations on rows starting with the powers of two. He then found "friendly pairs" or amicable numbers by neighboring numbers and corresponding primes.
Pierre de Fermat and Marin Mersenne discovered, in 1636, the amicable pair
17296 = 16*23*47 and 18416 = 16*1151
and Rene Descartes found the third pair
9363584 = 128*191*383 and 9437056 = 128*73727.
In 1747 Euler, as usual, went into overdrive and produced more amicable pairs than anyone had done before him. He published a paper "On the amicable numbers" adding 30 more pairs; and then three years he had extended the list to 60 amicable pairs. 16-year old Nicolo Paganini found, in 1866, another amicable pair which was missed by the great mathematicians before him: 1184 and 1210.
Euler's rule is a generalization of the Thâbit ibn Qurra theorem. It states that if
p = (2(n - m)+1) × 2m − 1,
q = (2(n - m)+1) × 2n − 1,
r = (2(n - m)+1)2 × 2m + n − 1,
where n > m > 0 are integers and p, q, and r are prime numbers, then 2n×p×q and 2n×r are a pair of amicable numbers.