Amicable Numbers

Number Theory · Axiom Academy

Exploring pairs of numbers united by a beautiful relationship of divisor sums 1. Definition of Amicable Numbers Two positive integers a and b are amicable if the sum of proper divisors of a equals b , and the sum of proper divisors of b equals a . Let σ(n) denote the sum of all divisors of n. Then a and b are amicable if: where σ(n) - n represents the sum of proper divisors (excluding n itself) Think of it as a mutual relationship: the divisors of the first number add up to the second, and the divisors of the second add up to the first. 2. The Classic Pair: 220 and 284 The smallest amicable pair, known since ancient times, consists of 220 and 284. Let's verify this relationship by computing the proper divisors of each number. Proper divisors: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110 Sum: 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284 Proper divisors: 1, 2, 4, 71, 142 Sum: 1 + 2 + 4 + 71 + 142 = 220 Notice the perfect symmetry: the proper divisors of 220 sum to 284, and the proper divisors of 284 sum to 220. This mutual relationship defines amicability. The story of amicable numbers spans millennia, from ancient mathematical texts to modern computer searches. Ancient Times: The Pythagoreans (circa 500 BCE) knew about 220 and 284, viewing them as symbols of friendship. They believed these numbers had mystical significance.

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