Definition:Perfect Number

Definition

A perfect number is a natural number equal to the sum of all its divisors except itself.

Formally stated, a perfect number $n$ is a natural number such that $\sigma \left({n}\right) = 2 n$, where $\sigma: \N \to \N$ is the sigma function.

It is an open question as to whether all perfect numbers are even. No odd perfect numbers have ever been found. It is known that no odd perfect number contains less than 100 digits.

An even perfect number is of the form $2^{n-1} \left({2^n - 1}\right)$ as proved here, where $2^n - 1$ is prime.

Examples

The first few perfect numbers are:

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 6\)	\(=\)	\(\displaystyle 2 \times 3\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 28\)	\(=\)	\(\displaystyle 2^2 \times 7\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 496\)	\(=\)	\(\displaystyle 2^4 \times 31\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 8128\)	\(=\)	\(\displaystyle 2^6 \times 127\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)

This sequence is A000396 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Euclid's Definition

As Euclid defined it:

A perfect number is that which is equal to the sum its own parts.

(The Elements: Book VII: Definition $22$)

Sources

• George E. Andrews: Number Theory (1971): $\S 3.5$