Odd Perfect Number — Definition, Formula & Examples

An odd perfect number is a hypothetical odd integer that equals the sum of all its proper divisors. No odd perfect number has ever been found, and whether one exists remains one of the oldest unsolved problems in mathematics.

An odd perfect number is an odd positive integer $n$ such that $\sigma(n) = 2n$ , where $\sigma(n)$ denotes the sum-of-divisors function. Equivalently, the sum of all positive divisors of $n$ less than $n$ equals $n$ itself, with $n$ being odd. As of today, no such number is known to exist, and it has been proven that if one does exist, it must exceed $10^{1500}$ .

Key Formula

\sigma(n) = 2n, \quad n \text{ odd}

Where:

$n$ = A hypothetical odd perfect number
$\sigma(n)$ = The sum of all positive divisors of n, including n itself

How It Works

A perfect number is one where the sum of its proper divisors (all divisors except itself) equals the number. For example,

6

is perfect because

1 + 2 + 3 = 6

. All known perfect numbers are even, and Euler proved that every even perfect number has the form

2^{p-1}(2^p - 1)

where

2^p - 1

is a Mersenne prime. The open question is whether any odd number can also be perfect. Mathematicians have not found one, but they also have not proven that none can exist. Researchers have established many necessary conditions: for instance, an odd perfect number must have at least 101 prime factors (counting multiplicity) and must be divisible by a prime raised to a power of the form

4k + 1

Worked Example

Problem: Verify that 6 is a perfect number but is even, and check whether 9 (an odd number) is perfect.

Step 1: Find the proper divisors of 6.

\text{Divisors of } 6: \; 1, 2, 3

Step 2: Sum the proper divisors of 6 and compare to 6.

1 + 2 + 3 = 6 \quad \checkmark \text{ (perfect)}

Step 3: Now find the proper divisors of 9 and check.

\text{Divisors of } 9: \; 1, 3 \quad \Rightarrow \quad 1 + 3 = 4 \neq 9

Answer: 6 is perfect but even. 9 is odd but not perfect, since its proper divisors sum to only 4. This illustrates why finding an odd perfect number is so difficult — odd numbers tend to have divisor sums that fall well short.

Why It Matters

The odd perfect number problem connects to deep structures in number theory studied in undergraduate and graduate mathematics. Working on necessary conditions for its existence has advanced techniques in modular arithmetic, prime factorization bounds, and computational number theory. It is often one of the first "easy to state, hard to solve" problems students encounter when exploring open questions in math.

Common Mistakes

Mistake: Assuming the problem is settled because no odd perfect number has been found.

Correction: "Not found" is not the same as "proven impossible." No proof of nonexistence is known. The problem remains open, meaning either outcome — existence or nonexistence — is still logically possible.