What Are Amicable Numbers?

Amicable numbers are a pair of two distinct positive integers $(a, b)$ such that the sum of the proper divisors (all divisors excluding the number itself) of $a$ equals $b$, and the sum of the proper divisors of $b$ equals $a$.

Example

• $(220, 284)$ is the smallest known pair of amicable numbers.
• Proper divisors of 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 → sum = 284
• Proper divisors of 284: 1, 2, 4, 71, 142 → sum = 220
• This property was already known in the ancient Pythagorean school.

Historical Discoveries

• Fermat: (17296, 18416)

• Descartes: (9363584, 9437056)

• Euler: discovered a total of 59 pairs of amicable numbers

• Escott: in 1866, published a paper describing 390 pairs

• Paganini: (1184, 1210), the second smallest pair of amicable numbers
  → Discovered by a 16-year-old Italian, and became widely known.

• To date, millions of amicable pairs have been discovered, but it is still unknown whether infinitely many exist.

• No pair of odd amicable numbers has ever been found, and their existence remains an open question.