166 7. ANCIENT NUMBER THEORY
into exactly one of these three classes. Such is not the case, however. The property
of primeness is a property of a number alone. The property of being relatively
prime is a property of a pair of numbers. On the other hand, the property of being
relatively prime to a given number is a property of a number alone. Nicomachus
explains the property in a rather wordy fashion in Chapter 13 of Book 1, where he
gives a method of identifying prime numbers that has become famous as the sieve
of Eratosthenes.
Nicomachus attributes this method to Eratosthenes. To use it, start with a list
of all the odd numbers from 3 on, that is,
3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37,....
From this list, remove every third number after 3, that is, remove 9, 15, 21, 27,
33,.... These numbers are multiples of 3 and hence not prime. The reduced list
is then
3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 47, 49,....
From this list, remove all multiples of 5 larger than 5. The first non-prime in the
new list is 49 = 7 · 7. In this way, you can generate in short order a complete list
of primes up to the square of the first prime whose multiples were not removed.
Thus, after removing the multiples of 7, we have the list
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61....
The first non-prime in this list would be 11 • 11 = 121.
Nicomachus' point of view on this sieve was different from ours. Where we
think of the factors of, say 60, as being 2, 2, 3, and 5, Nicomachus thought of the
quotients by these numbers and products of them as the parts of a number. Thus,
in his language 60 has the parts 30 (half of 60), 20 (one-third of 60), 15 (one-fourth
of 60), 12 (one-fifth of 60), 10 (one-sixth of 60), 6 (one-tenth) of 60, 5 (one-twelfth
of 60), 4 (one-fifteenth of 60), 3 (one-twentieth of 60), 2 (one-thirtieth of 60) and
1 (one-sixtieth of 60). If these parts are added, the sum is 108, much larger than
- Nicomachus called such a number superabundant and compared it to an animal
having too many limbs. On the other hand, 14 is larger than the sum of its parts.
Indeed, it has only the parts 7, 2, and 1, which total 10. Nicomachus called 14 a
deficient number and compared it to an animal with missing limbs like the one-eyed
Cyclops of the Odyssey. A number that is exactly equal to the sum of its parts,
such as6 = l + 2 + 3, he called a perfect number. He gave a method of finding
perfect numbers, which remains to this day the only way known to generate such
numbers, although it has not been proved that there are no other such numbers.
This procedure is also stated by Euclid: // the sum of the numbers 1, 2, 4,..., 2 n_1
is prime, then this sum multiplied by the last term will be perfect. The modern
statement of this fact is given in the exercises below. To see the recipe at work,
start with 1, then double and add: 1 + 2 = 3. Since 3 is prime, multiply it by the
last term, that is, 2. The result is 6, a perfect number. Continuing, 1 + 2 + 4 = 7,
which is prime. Multiplying 7 by 4 yields 28, the next perfect number. Then,
1 + 2 + 4 + 8 + 16 = 31, which is prime. Hence 31 - 16 = 496 is a perfect number.
The next such number is 8128 = 64(1 + 2 + 4 + 8+ 16 + 32 + 64). In this way,
Nicomachus was able to generate the first four perfect numbers. He seems to hint
at a conjecture, but draws back from stating it explicitly:
