amicable numbers

amicable numbers, in mathematics, a pair of integers in which each is the sum of the divisors of the other. The first pair of amicable (“friendly”) numbers, 220 and 284, was discovered by the ancient Greeks. The sum of the proper divisors of 284 is 1 + 2 + 4 + 71 + 142 = 220, and the sum of the proper divisors of 220 is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284. (Iamblichus attributed this discovery to Pythagoras, but modern historians doubt this claim.)

For centuries, 220 and 284 were the only known amicable pair. With the rise of learning in the Islamic world, scholars there absorbed the works of other civilizations and augmented these with homegrown achievements. Thābit ibn Qurrah (active in Baghdad in the 9th century) returned to the Greek problem of amicable numbers and discovered a rule for generating them. If three numbers x, y, and z are primes where x = 3 × 2ⁿ⁻¹ − 1, y = 3 × 2ⁿ − 1, and z = 9 × 2²ⁿ⁻¹ − 1, then the numbers A = 2ⁿxy and B = 2ⁿz are an amicable pair. For n = 2, this rule generates 220 and 284. Five hundred years later the second pair, 17,296 and 18,416, was found for n = 4, and three centuries after that a third pair, 9,363,584 and 9,437,056, was found for n = 7.

French mathematicians Pierre de Fermat and René Descartes independently rediscovered Thābit ibn Qurrah’s rule in the 17th century and found the second and third pairs, respectively. In the 18th century, Swiss mathematician Leonhard Euler came up with new rules for such numbers and discovered 59 pairs. In 1866, a 16-year-old Italian schoolboy, Nicoló Paganini (not to be confused with the violinist), discovered the second smallest amicable pair, 1,184 and 1,210, which had been overlooked by previous mathematicians.

With the introduction of computers, amicable numbers have been found much faster. As of the mid-20th century, 390 pairs were known; as of 2023, 1,227,869,886 pairs are known. A distributed computing project has found all pairs smaller than 10²⁰ (one hundred billion billion). Amicable pairs are rare, and it is not known if there are infinitely many amicable numbers.