Amicable Numbers

Amicable numbers are pairs of numbers that have the property that each of the two numbers is the sum of the proper divisors of the other. 220 and 284 form the smallest pair of amicable numbers. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, which sum to 284. The proper divisors of 284 are 1, 2, 4, 71, and 142, which total 220. This pair has been known to be amicable at least since the time of Pythagoras. The second smallest pair, 1184 and 1210, was discovered as recently as 1866 by Nicolo Paganini when he was a 16-year-old schoolboy (although this pair may have been previously known to Arab mathematicians).

Thabit ibn Qurra, a brilliant Arab mathematician, astronomer and physician, described in his Book on the Determination of Amicable Numbers Euclid's rule for perfect numbers (Elements, Book IX, proposition 36) and gave a formula for obtaining pairs of amicable numbers.

Thabit's formula involves finding a number n, greater than 1, that makes these three expressions all prime:
a = 3 × 2ⁿ − 1
b = 3 × 2^n-1 − 1
c = 9 × 2^2n-1 − 1
The two numbers 2ⁿ × a × b and 2ⁿ × c will be amicable. Such pairs of numbers are called Thabit pairs. For n = 2, the pair is 220 and 284. For n = 4, the formula gives 17,296 and 18,416. For n = 7, the formula produces 9,363,584 and 9,437,056. However, it is quite difficult to get a, b, and c all prime simultaneously. n = 2, 4, or 7 are the only values found so far that make all three numbers prime.

Thabit's rule does not give all pairs of amicable numbers. Rather, it is one of a number of similar patterns that generate amicable pairs. There is no one rule to generate all amicable pairs, as there is to generate all (even) perfect numbers.

Hundreds of thousands of pairs of amicable numbers are now known. Most of these pairs have been discovered only in the past few years.

Here are all amicable pairs less than 100,000:

You may also be interested in reading about:

• Sociable numbers
• Perfect numbers
• Aliquot chains
• Loops

See also Amicable numbers from "The World of Trotter Math". This page contains interesting information and several useful links related to amicable numbers.