We will learn what a perfect number is in simple terms. This includes its definition, important formulas, and a list of perfect numbers from 1 to 100. We’ll also understand Euclid’s perfect number rule. A few solved examples are included to help you prepare better for exams. You’ll also learn the steps to find perfect numbers on your own.

What is a Perfect Number?

A perfect number is a whole number that is equal to the sum of all its proper divisors (divisors that are smaller than the number itself).

For example:

• The number 6 has divisors 1, 2, and 3.
• If you add them: 1 + 2 + 3 = 6, which is the number itself.
  So, 6 is a perfect number.

Other examples of perfect numbers are 28, 496, and 8128.

A perfect number is a positive number whose proper divisors (excluding the number itself) add up to the number.

For example, the divisors of 6 (excluding 6) are 1, 2, and 3. Since 1 + 2 + 3 = 6, it is a perfect number.

In math terms, a number N is perfect if the sum of all its divisors (including itself), written as σ(N), is equal to 2N.

If σ(N) < 2N, it's called deficient; if σ(N) > 2N, it's called abundant.

The definition of perfect number is equivalent to saying that the sum of the proper (or aliquot) divisors of N is equal to N (we just do not add N itself to the sum). While this may seem more natural, the central reason for using the function σ is that it possesses some very special properties.

History of Perfect Numbers 

Perfect numbers have been known since ancient times, although no one knows exactly who discovered them first. It's believed that even the Egyptians might have known about them long ago.

The Greek mathematician Pythagoras and his followers were especially interested in perfect numbers. They thought the number 6 was special because it equals the sum of its divisors (except itself):
6 = 1 + 2 + 3
This made it the smallest perfect number. The next one is 28.

At that time, people were more focused on the mystical or magical meanings of numbers rather than deep math. But real progress came around 300 BC, when Euclid, a famous Greek mathematician, wrote a book called "Elements." Even though the book is about geometry, it also contains some early ideas about number theory, including perfect numbers.

In simple terms, a perfect number is a positive whole number that is equal to the sum of its proper divisors (all divisors except the number itself). For example, 6 is perfect because 1 + 2 + 3 = 6.

Euclid’s Perfect Number Theorem

The Euclid–Euler Theorem explains how perfect numbers are related to a special kind of prime number called a Mersenne prime.

A Mersenne prime is a number in the form 2ᵖ − 1, where p is a prime number and the result is also prime.
The theorem says:

Every even perfect number can be written as 2^(p−1) × (2ᵖ − 1), where 2ᵖ − 1 is a Mersenne prime.

This means if you find a Mersenne prime, you can use this formula to get a perfect number.

Examples using the formula:

• If p = 2:
  2¹ × (2² − 1) = 2 × 3 = 6
• If p = 3:
  2² × (2³ − 1) = 4 × 7 = 28
• If p = 5:
  2⁴ × (2⁵ − 1) = 16 × 31 = 496
• If p = 7:
  2⁶ × (2⁷ − 1) = 64 × 127 = 8128

Perfect Number Example

6 = 3 + 2 + 1, for example, and 28 = 14 + 7 + 4 + 2 + 1.

A number is perfect if the sum of its appropriate components equals the number.

Make a list of all the numbers that divide a number except the number itself to find the perfect factors.

The numbers 1, 2, 3, 6, and 9 are ideal factors for the number 18.

If the sum of the elements equals 18, it is a perfect number.

1 + 2 + 3 + 6 + 9 = 6 + 6 + 9 = 12 + 9 = 21

As a result, 18 isn’t perfect.

Perfect Numbers Table

A Perfect Numbers Table lists special positive integers that are equal to the sum of their proper divisors (excluding the number itself). These numbers are often generated using a formula involving Mersenne primes, and include values like 6, 28, 496, and so on.

Perfect Numbers List from 1 to 100

There are only two perfect numbers from 1 to 100. They are 6 and 28.

The first 10 perfect numbers are as follows:

1. 6
2. 28
3. 496
4. 8128
5. 33550336
6. 8589869056
7. 137438691328
8. 2305843008139952128
9. 2658455991569831744654692615953842176
10. 191561942608236107294793378084303638130997321548169216

How to Find the Perfect Number?

We can easily find perfect numbers using the following steps:

Perfect numbers are positive integers n such that n=s(n),

where, s(n) is the restricted divisor function (i.e., the sum of proper divisors of n), or equivalently

sigma(n)=2n,

where sigma(n) is the divisor function (i.e., the sum of divisors of n including n itself). For example, the first few perfect numbers are 6, 28, 496, 8128, … (OEIS A000396), since

6 = 1+2+3

28 = 1+2+4+7+14

496 = 1+2+4+8+16+31+62+124+248.

Important Notes on Perfect Numbers

• Perfect numbers are special positive whole numbers where the sum of all their proper divisors (factors except the number itself) is equal to the number.
• The smallest perfect number is 6, because 1 + 2 + 3 = 6.
• So far, all known perfect numbers are even (divisible by 2). No odd perfect number has ever been found.
• Interestingly, perfect numbers usually end with 6 and 8 one after the other (like 6, 28, 496, 8128, etc.).

Perfect Numbers Solved Examples

Example 1: Find out if 496 is a perfect number.

Solution: The proper factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, and 248

1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

496 is perfect.

Example 2: Is 6 a perfect number?

Solution:
Find all divisors of 6 except 6:
1, 2, 3

Add them:
1 + 2 + 3 = 6

Since the sum equals the number itself,
6 is a perfect number.

Example 3: Is 28 a perfect number?

Solution:
Divisors of 28 excluding 28:
1, 2, 4, 7, 14

Add them:
1 + 2 + 4 + 7 + 14 = 28

So, 28 is a perfect number.

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