A perfect number is a positive integer the sum of whose proper divisorsA proper divisor is a number n that when divided into another number N leaves no remainder. is equal to the number itself. The smallest perfect number is 6, which has proper divisors of 1, 2, and 3.[a] The next three perfect numbers, in ascending order, are 28, 496, and 8128, all of which were known to ancient mathematicians.[1] All known perfect numbers are even; it is uncertain whether there are any odd perfect numbers.[2]

Followers of the Ancient Greek philosopher Pythagoras had a particular interest in the mystical properties of numbers.[2] Pythagoras is credited with coining the term perfect to describe the numbers 6 and 28, because they correspond to heavenly events: the number of days it took God to create the world and the number of days it takes the Moon to complete an orbit of the Earth, the lunar cycle.[3][b] In the words of St Augustine, writing in De civitate Dei contra paganos (The City of God) in the early 5th century:

Six is a number perfect in itself, and not because God created all things in six days; rather, the converse is true. God created all things in six days because the number is perfect.



Ben-Menaḥem, Ari. (2009). Historical Encyclopedia of Natural and Mathematical Sciences (Vol. 2). Springer Science & Business Media.
Stakhov, A. (2009). The Mathematics of Harmony: From Euclid to Contemporary Mathematics and Computer Science. World Scientific Publishing.
Weisstein, E. W. (n.d.). Perfect number. MathWorld. http://mathworld.wolfram.com/PerfectNumber.html


  1. The list of proper divisors of any positive integer includes 1, which is a proper divisor of every number, but excludes the integer itself.
  2. It takes the Moon 27.3 days to complete a revolution of the Earth, but because the Earth itself is orbiting around the Sun it takes a little longer for the Moon to appear to an observer on Earth to be in the same position relative to the Sun as it was at the start of the month.