Numbers: Their Tales, Types, and Treasures.
Chapter 9: Number Relationships
9.2.AMICABLE NUMBERS
What could possibly make two numbers amicable, or friendly? The friendliness of these two numbers will be shown in their “amicable” relationship to each other. This relationship is defined in terms of the proper divisors of these numbers. A proper divisor of a natural number n is any natural number that divides n, excluding n itself. For example, 12 has the proper divisors 1, 2, 3, 4, and 6 (but not 12). A number is called perfect if the sum of its proper divisors is equal to the number (see chapter 8, section 5). Two numbers are considered amicable if the sum of the proper divisors of one number equals the second number and the sum of the proper divisors of the second number equals the first number. Perhaps the best way to understand this is through an example. Let's look at the smallest pair of amicable numbers: 220 and 284.
· The divisors of 220 (other than 220 itself) are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110.
· Their sum is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.
· The divisors of 284 (other than 284 itself) are 1, 2, 4, 71, and 142, and their sum is 1 + 2 + 4 + 71 + 142 = 220.
This shows that the two numbers are amicable numbers.
This pair of amicable numbers was already known to Pythagoras by about 500 BCE.
A second pair of amicable numbers is 17,296 and 18,416. The discovery of this pair is usually attributed to the French mathematician Pierre de Fermat (1607–1665), although there is evidence that this discovery was anticipated by the Moroccan mathematician Ibn al-Banna al-Marrakushi al-Azdi (1256–ca. 1321).
The sum of the proper divisors of 17,296 is
1 + 2 + 4 + 8 + 16 + 23 + 46 + 47 + 92 + 94 + 184 + 188 + 368 + 376 + 752 + 1,081 + 2,162 + 4,324 + 8,648 = 18,416.
The sum of the proper divisors of 18,416 is
1 + 2 + 4 + 8 + 16 + 1,151 + 2,302 + 4,604 + 9,208 = 17,296.
Thus, they, too, are truly amicable numbers!
French mathematician René Descartes (1596–1650) discovered another pair of amicable numbers: 9,363,584, and 9,437,056. By 1747, Swiss mathematician Leonhard Euler (1707–1783) had discovered sixty pairs of amicable numbers, yet he seemed to have overlooked the second-smallest pair—1,184 and 1,210, which were discovered in 1866 by the sixteen-year-old B. Nicolò I. Paganini.
The sum of the divisors of 1,184 is
1 + 2 + 4 + 8 + 16 + 32 + 37 + 74 + 148 + 296 + 592 = 1,210.
And the sum of the divisors of 1,210 is
1 + 2 + 5 + 10 + 11 + 22 + 55 + 110 + 121 + 242 + 605 = 1,184.
To date we have identified over 363,000 pairs of amicable numbers, yet we do not know if there are an infinite number of such pairs. The table in the appendix, section 7, provides a list of the first 108 amicable numbers. An ambitious reader might want to verify the “friendliness” of each of these pairs. Going beyond this list, we will eventually stumble on an even larger pair of amicable numbers: 111,448,537,712 and 118,853,793,424.
Readers who wish to pursue a search for additional amicable numbers might want to use the following method for finding them: Consider the numbers
a = 3 × 2n − 1, b = 3 × 2n − 1 − 1, and c = 32 × 22n − 1 − 1,
where n is an integer ≥ 2.
If a, b, and c are prime numbers, then 2n × a × b and 2n × c are amicable numbers. (For n ≤ 200, only n = 2, 4, and 7 would give us a, b, and c to be prime numbers.)
Inspecting the list of amicable numbers in the appendix, section 7, we notice that each pair is either a pair of odd numbers or a pair of even numbers. To date, we do not know if there is a pair of amicable numbers where one is odd and one is even. We also do not know if any pair of amicable numbers is relatively prime (that is, the numbers have no common factor other than 1). These open questions contribute to our continued fascination with amicable numbers.