What Is A Perfect Number? (3 Key Concepts You Should Know)

Perfect numbers are an interesting concept in number theory.  To understand them, we need to find the divisors of an integer, take their sum, and see how that sum compares to the integer.

So, what is a perfect number?  A perfect number N is a positive integer that is equal to the sum of all positive divisors of N that are less than N. The smallest perfect number is 6 (6 = 1 + 2 + 3). All known perfect numbers are even, and it is unknown if there are any odd perfect numbers. A prime number cannot be perfect.

Of course, once numbers become large enough, it is only practical to check if they are perfect by using a computer.

In this article, we’ll talk about what perfect numbers are and how to find or verify them.  We’ll also answer some common questions about perfect numbers.

Let’s get started.

What Is A Perfect Number?

A perfect number N is a positive integer (a whole number with no decimals or fractions) that is equal to the sum of all its positive divisors (factors) that are less than N.

For example, the number 6 is a perfect number because:

• 6 is a positive integer.
• The positive divisors of 6 are 1, 2, 3, and 6.
• The sum of the positive divisors of 6 that are less than 6 is 1 + 2 + 3 = 6

Another perfect number is 28, since:

• 28 is a positive integer.
• The positive divisors of 28 are 1, 2, 4, 7, 14, and 28.
• The sum of the positive divisors of 28 that are less than 28 is 1 + 2 + 4 + 7 + 14 = 28.

The first few integers in the perfect number sequence are 6, 28, 496, and 8128.

How To Find Perfect Numbers (Calculate Perfect Numbers)

We can use a more general equation to figure out if a number N is perfect.

Let d_i be the positive divisors of the number N, in increasing order.  Then d₁ < d₂ < … < d_M, and d₁, d₂, … , d_M are all divisors of N.

Also, d₁ = 1, since 1 is the smallest divisor of N, and d_M = N, since N is the largest divisor of N.

If we take the sum of all the positive divisors of N, we get S = Σd_i.

To exclude d_M = N, we take S – N = Σd_i – N.  This gives us the sum of all positive divisors of N that are less than N.

If N = S – N (or S = 2N), then N is perfect.  Note that S = 2N means that we get an even value when we take the sum all positive divisors of N that are less than N.

Let’s look at some examples to see how we can tell if a number is perfect or not.

Example 1: Is 2 A Perfect Number?

Since 2 is prime, it is not a perfect number (you can see the proof later in this article).

However, we can also see this by listing out factors:

• The positive divisors of 2 are 1 and 2.
• The sum of the positive divisors of 2 that are less than 2 is 1.

Since 1 is not equal to 2, 2 is not a perfect number.

Example 2: Is 3 A Perfect Number?

Since 3 is prime, it is not a perfect number (you can see the proof later in this article).

However, we can also see this by listing out factors:

• The positive divisors of 3 are 1 and 3.
• The sum of the positive divisors of 3 that are less than 3 is 1.

Since 1 is not equal to 3, 3 is not a perfect number.

Example 3: Is 4 A Perfect Number?

4 is not a perfect number, and we can see this by listing out factors:

• The positive divisors of 4 are 1, 2, and 4.
• The sum of the positive divisors of 4 that are less than 4 is 1 + 2 = 3.

Since 3 is not equal to 4, 4 is not a perfect number.

Example 4: Is 5 A Perfect Number?

Since 5 is prime, it is not a perfect number (you can see the proof later in this article).

However, we can also see this by listing out factors:

• The positive divisors of 5 are 1 and 5.
• The sum of the positive divisors of 5 that are less than 5 is 1.

Since 1 is not equal to 5, 5 is not a perfect number.

Example 5: Is 6 A Perfect Number?

6 is a perfect number, and we can see this by listing out factors:

• The positive divisors of 6 are 1, 2, 3, and 6.
• The sum of the positive divisors of 6 that are less than 6 is 1 + 2 + 3 = 6.

Since 6 is equal to 6, 6 is a perfect number.

Example 6: Is 7 A Perfect Number?

Since 7 is prime, it is not a perfect number (you can see the proof later in this article).

However, we can also see this by listing out factors:

• The positive divisors of 7 are 1 and 7.
• The sum of the positive divisors of 7 that are less than 7 is 1.

Since 1 is not equal to 7, 7 is not a perfect number.

Example 7: Is 8 A Perfect Number?

8 is not a perfect number, and we can see this by listing out factors:

• The positive divisors of 8 are 1, 2, 4, and 8.
• The sum of the positive divisors of 8 that are less than 8 is 1 + 2 + 4 = 7.

Since 7 is not equal to 8, 8 is not a perfect number.

Example 8: Is 9 A Perfect Number?

9 is not a perfect number, and we can see this by listing out factors:

• The positive divisors of 9 are 1, 3, and 9.
• The sum of the positive divisors of 9 that are less than 9 is 1 + 3 = 4.

Since 4 is not equal to 9, 9 is not a perfect number.

Example 9: Is 10 A Perfect Number?

10 is not a perfect number, and we can see this by listing out factors:

• The positive divisors of 10 are 1, 2, 5, and 10.
• The sum of the positive divisors of 10 that are less than 10 is 1 + 2 + 5 = 8.

Since 8 is not equal to 10, 10 is not a perfect number.

Can A Perfect Number Be Odd?

There are no known odd perfect numbers – all known perfect numbers are even.  However, it is possible that there is an odd perfect number that we have not discovered yet.

It is yet to be proven (or disproven) that a perfect number must be odd.

Can Perfect Numbers Be Negative?

Perfect numbers cannot be negative.  By definition, a perfect number N is a positive integer.

The reason is that we are only considering the positive divisors of N.  If we considered both positive and negative divisors of an integer N, then the sum of all the divisors would always be zero (because for each positive divisor d_i, there is a negative divisor –d_i).

Can A Prime Number Be A Perfect Number?

A prime number cannot be a perfect number. This means that any perfect number is composite.  We can prove this as follows.

Let N be a prime number.  Then the only positive divisors of N are 1 and N (by definition of a prime number).

Then the only positive divisor of N that is less than N is 1.  So, the sum of all positive divisors of N that are less than N is just 1.

For any prime number N, 1 is never equal to N (since the smallest prime is 2).  So, a prime number N cannot be perfect.

How Many Perfect Numbers Are There?

There are 51 known perfect numbers, all of which are even.  However, there may be more perfect numbers that have not yet been discovered  and verified.

It is unknown if there are any odd perfect numbers.  It is also unknown whether there are infinitely many even perfect numbers.  Both of these still need to be proved or disproved.

You can find a list of known perfect numbers on Wikipedia.

After the first four perfect numbers (6, 28, 496, and 8,128), the remaining perfect numbers are 8 digits or larger (the fifth perfect number is 33,550,336).

What Is The Largest Perfect Number?

The largest known perfect number is incredibly large: it has 49,724,095 digits.  If you tried to write it out by hand and wrote 5 digits per second, it would take you 288 days of nonstop writing!

Conclusion

Now you know what a perfect number is and how you can find them (or at least verify them).  You also know what can and cannot be perfect numbers.

I hope you found this article helpful.  If so, please share it with someone who can use the information.

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~Jonathon