Plato's Numbers

The positive integers 216 and  appear in an obscure passage in Plato's The Republic. In this passage, Plato alludes to the fact that 216 is equal to , where 6 is one of the numbers representing marriage since it is the product to the female 2 and the male 3. Plato was also aware of the fact the sum of the cubes of the 3-4-5 Pythagorean triple is equal to  (Livio 2002, p. 66).

In Laws, Plato suggests that  is the optimal number of citizens in a state because

1. It is the product of 12, 20, and 21.

2. The 12th part of it can still be divided by 12.

3. It has 59 proper divisors, including all numbers for 1 to 12 except 11, and 5038--which is very close to 5040--is divisible by 11

(Livio 2002, p. 65).

REFERENCES:

Heath, T. L. Aristarchus of Samos: The Ancient Copernicus. New York: Dover, pp. 171-172, 1981.

Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number. New York: Broadway Books, pp. 65-66, 2002.

Plato. The Republic. New York: Oxford University Press, 1994.

Wells, D. G. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin, p. 144, 1986.