A perfect number is a positive integer that equals the sum of its proper divisors, that is, positive divisors excluding the number itself. For example, is a perfect number because the proper divisors of are and and
The sum of all positive divisors of a number is denoted by . A perfect number is therefore a positive integer such that
Perfect numbers were of great interest to ancient mathematicians including the Greeks, who attributed mystical significance to the property. It is perhaps somewhat surprising that many elementary questions about perfect numbers are still open. Most notably, it is not known whether there are infinitely many perfect numbers, and it is not known whether there are any odd perfect numbers.
Euclid proved in the Elements that if is prime, then is perfect.
The first four perfect numbers are
More than 2000 years later, Euler was the first to give a proof that every even perfect number was of this form. This is known as the Euclid-Euler theorem. Euler's proof is quite elementary:
A positive integer is an even perfect number if and only if for some positive prime such that is prime. Here is known as a Mersenne prime.
For the "if" (Euclid) statement, let be the Mersenne prime , and write down the divisors of explicitly: they are and . The sum of these divisors, by the geometric series formula, is
So the sum of the proper divisors is .
For the "only if" (Euler) statement, some facts about the sum of divisors function, denoted , will be relevant. In particular, is multiplicative: if are relatively prime.
Now suppose is an even perfect number. Then write where and is odd. Since is perfect, . Then
So , hence , hence for some integer , and expanding the above equation again yields
But note that the right side is at least since these are two divisors of , and those two divisors already add up to . So the conclusion is that equality holds and these are the only two divisors of . This can only happen if and is prime. This completes the proof.
Exercise: Where did the proof use the fact that
The following is a list of some interesting properties of perfect numbers. Unlike many lists in the literature, this list is confined to nontrivial properties: for instance, the fact that every even perfect number is a triangular number is a trivial consequence of the formula
- Every even perfect number ends in the digits or (when written in decimal).
- The iterative digital sum of any even perfect number except is .
- The only square-free perfect number is .
- A perfect number is an Ore harmonic number; that is, the harmonic mean of its divisors is an integer. (Not every Ore harmonic number is perfect, e.g. 140.)
- If an odd perfect number exists, it has more than digits, at least prime factors, at least distinct prime factors, and at least one prime factor of at least digits.
Proofs: Here are hints for 1-4:
This follows from a mod-10 and mod-100 analysis. If mod or , the corresponding perfect number ends in , and if mod , it ends in hint: show that it is congruent to mod
Show that every even perfect number except is mod . Summing the digits and iterating preserves the congruence class mod .
For even perfect numbers this is clear from Euclid-Euler; for odd perfect numbers, two odd prime factors would lead to a factor of in , but isn't divisible by .
The harmonic mean of the divisors of a perfect number is (why?), where is the number of divisors. Now is even unless is a square, but a perfect square cannot be a perfect number because the sum of its divisors is odd (why?).
This represents the current frontier of mathematical research on odd perfect numbers. The methods used to establish these results seem exceedingly unlikely to lead to a proof that odd perfect numbers do not exist since they provide lower bounds only. Perhaps the existence of an odd perfect number seems unlikely given 5, but sometimes interesting phenomena only happen for large enough integers.