Fermat's Enigma

by Simon Singh

One of Fermat’s discoveries concerned the so-called friendly numbers, or amicable numbers, closely related to the perfect numbers that had fascinated Pythagoras two thousand years earlier. Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number. The Pythagoreans made the extraordinary discovery that 220 and 284 are friendly numbers. The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, and the sum of all these is 284. On the other hand, the divisors of 284 are 1, 2, 4, 71, 142, and the sum of all these is 220. The pair 220 and 284 was said to be symbolic of friendship. Martin Gardner’s book Mathematical Magic Show tells of talismans sold in the Middle Ages that were inscribed with these numbers on the grounds that wearing the charms would promote love. An Arab numerologist documents the practice of carving 220 on one fruit and 284 on another, and then eating the first one and offering the second one to a lover as a form of mathematical aphrodisiac. Early theologians noted that in Genesis Jacob gave 220 goats to Esau. They believed that the number of goats, one half of a friendly pair, was an expression of Jacob’s love for Esau.