Numbers measured by their parts
Greek arithmetic spent much effort on how numbers relate to their divisors. A proper divisor of a number is a divisor smaller than the number itself. Adding up the proper divisors gives a total that may be smaller than, larger than, or equal to the number. This simple comparison produced a classification and some of the most durable puzzles in number theory.
Perfect numbers
A number is perfect if it equals the sum of its proper divisors. The smallest is 6, whose proper divisors 1, 2 and 3 add up to 6. The next is 28, since 1 plus 2 plus 4 plus 7 plus 14 is 28. After that come 496 and 8128. For 496, the proper divisors 1, 2, 4, 8, 16, 31, 62, 124 and 248 add to 496. The reputation of the perfect number six in Pythagorean circles belongs to a wider tradition of number symbolism; here the point is the arithmetic.
What Euclid proved
In Book IX of the Elements, around 300 BCE, Euclid proved a rule for building perfect numbers. In modern notation: if 2 raised to the power n, minus 1, is a prime number, then 2 raised to the power n minus 1, multiplied by that prime, is perfect. Try n equal to 2: 3 is prime, so 2 times 3, which is 6, is perfect. For n equal to 3, 7 is prime, giving 4 times 7, which is 28. For n equal to 5, 31 is prime, giving 16 times 31, which is 496. For n equal to 7, 127 is prime, giving 64 times 127, which is 8128.
The result depends on the properties of prime numbers, and Euclid’s treatment of primes in the same book makes it a natural companion to that theory. He did not prove the reverse, that every perfect number has this form. Leonhard Euler did so in the eighteenth century for even perfect numbers. Whether an odd perfect number exists is still an open question, and none is known.
Nicomachus and the moral arithmetic of divisors
Nicomachus of Gerasa, writing his Introduction to Arithmetic around 100 CE, popularised the classification of numbers by their divisors. A number whose proper divisors add to more than itself he called overabundant, one whose divisors add to less he called deficient, and the rare case of equality perfect. He described these in moral terms: abundant numbers were like a monster with too many limbs, deficient ones like a creature missing some, and perfect numbers were a mean between excess and lack, comparable to virtue.
That sort of language reflects a view in which numbers carried ethical meaning, and later writers repeated it. It should be read as a way of thinking about number in that school, not as a property of the numbers themselves.
Nicomachus also made claims that turned out to be wrong. He suggested that there is exactly one perfect number in each range of numbers from one power of ten to the next, and that perfect numbers end alternately in 6 and 8. The first four do end 6, 8, 6, 8, but the fifth, 33,550,336, ends in 6 again, and perfect numbers turn out to be far scarcer than one per decade. This is a useful reminder that ancient authors generalised from the few cases they knew.
The pair 220 and 284
Amicable numbers are two numbers each of which equals the sum of the other’s proper divisors. The classical example is 220 and 284.
- The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Their sum is 284.
- The proper divisors of 284 are 1, 2, 4, 71 and 142. Their sum is 220.
Later writers in the Pythagorean tradition, such as Iamblichus in late antiquity, report that Pythagoras was asked what a friend is and answered “another I”, pointing to the pair 220 and 284 as the arithmetical image of friendship. That report comes many centuries after Pythagoras, and the attribution should be treated as tradition, not as evidence about the historical Pythagoras. It is, however, a clear example of how number properties were given social meaning.
Some writers in the medieval period also described talismans using the pair, and there are claims about its use in love charms. These claims are reported by later sources and are best taken as beliefs people held, with no effect implied.
After the Greeks
Interest in the pair continued. In the ninth century, the scholar Thabit ibn Qurra, working in Baghdad, gave a rule for generating amicable pairs. Pierre de Fermat found the pair 17296 and 18416 in 1636, and Descartes found another soon after. Euler later produced a long list of pairs, and in 1866 a sixteen-year-old Italian, Nicolo Paganini, noticed the small pair 1184 and 1210, which had escaped earlier searchers. Modern computers have since found a very large number of pairs.
Why the topic mattered to the Greeks
For Greek thinkers, these classifications served a larger goal: understanding number as an ordered structure with kinds and relations, something like a natural history of the integers. Perfect and amicable numbers are rare, which made them seem special. Their rarity is also the reason the questions remain interesting to mathematicians today, though for reasons of pattern rather than symbolism.