**Perfect Numbers:** A perfect number can be defined as an whole number (non-zero). The number that can be written as the sum of all its proper divisors is called a perfect number. Proper divisors include all the divisors of a number except the number itself. If the number itself is included then the perfect number will be half the sum of all its divisors.

All perfect numbers end in six or eight to be an alternating pattern for the first few perfect numbers. All are of the form ** 2 ^{n−1}(2^{n} − 1), ** where 2

^{n}-1 is a mersenne prime, so that the search for perfect numbers is the search for mersenne primes.

Result is applicable only for the even perfect numbers, no odd perfect number had been found among all the numbers till date.

A perfect number is a number equal to the sum of all of its proper divisors.

**Highlights of Perfect Numbers:**

- The total number of perfect numbers is not found. Since, the number of perfect numbers is proportional to the number of prime numbers.
- There are 37 known prime numbers and perfect numbers are found.
- The perfect numbers are not an odd number.

Example : 28 is a perfect number ?

Step1: Divisors of 28 are 1, 2, 4, 7, 14 and 28

Step 2: Sum of all proper divisor of 28 is

1 + 2 + 4 + 7 + 14 = 28

So 28 is a perfect number.

## List of Perfect Numbers

The following table shows the **list of perfect numbers:**

Perfect Number | Sum of Divisors |
---|---|

6 | 1 + 2 + 3 |

28 | 1 + 2 + 4 + 7 + 14 |

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

8128 | 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 |

## How to Find Perfect Numbers

The formula for this method is given by

**Example:**

3 can be given by = 2 × 2 – 1 = factors of 3: 1, 2.

7 can be given by = 4 × 2 – 1 = factors of 7: 1, 2, 4.

31 can be given by = 16 × 2 – 1 = factors of 31: 1, 2, 4, 8, 16.

127 can be given by = 64 × 2 – 1 = factors of 127: 1, 2, 4, 8, 16, 32, 64.

8191 can be given by = 4096 × 2 – 1 = factors of 8191: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096.

### Aliter Method

Perfect Number of a number can be obtained by using the formula.

**2**

^{n−1}x (2^{n}− 1)**Example:**

6 = > 2 × 3.

28 => 4 × 7.

496 =>16 × 31

8,128 => 64 × 127.

33, 550, 336 => 4096 × 8191.

### Calculate a perfect number from a Mersenne prime

To calculate a perfect number from a Mersenne prime number.

Let us consider ‘P’ is a prime number.

P = (2^{n} – 1).

The factors of 2^{(n-1)} are 1, 2, 4, 8, 16, 32, 64, 128….. 2^{(n-3)}, 2^{(n-2)} & 2^{(n-1)}.

The rest of the factors are multiplied by the above factors by 2^{(n-1)}.

The sum of L.H.S. is equal to [1+ 2 + 4 + 8 + 16 + … + 2^{(n-2)}].

On multiplying L.H.S by P we have P[1+ 2 + 4 + 8 + 16 + … + 2^{(n-2)}].

The sum of the R.H.S. is also found by using the distributive property of addition.

The sum of the R.H.S. is equal to 2^{n-1} (2^{n} – 1) – (2^{n} – 1) = [2^{(n-1)} – 1 + 1] × (2^{n} – 1)

= [2^{(n-1)} ] × (2^{n} – 1).

### By addition

1 × (2^{n} – 1) + [2^{(n-1)} – 1] × (2^{n} – 1) = [2^{(n-1)}] × (2^{n} – 1).

** If (2 ^{n} – 1) is a prime number, 2^{(n-1)} **

Thus, the perfect number is calculated from the mersenne prime number.

## Odd Perfect Numbers

There are no odd perfect number has been found up to 10^{300}. It is guessed that there are no odd perfect numbers. If there are some, then they are quite large over 300 digits and have numerous prime factors and in the form of P_{1}^{2e1}_{. }P_{2} ^{2e2} ………. ….P_{n}^{2en} _{ ,} Where P_{1} , P_{2} , …… , P_{n} are the distinct prime number.

Perfect Number of a number can be obtained by using the formula 2^{n – 1} × (2^{n} – 1), where 2^{n} – 1 is a prime number but It is not known if there are any odd perfect numbers.

## Examples of Perfect Numbers

**Question 1: Is 6 is a Perfect number?**

Solution:

The factors of 6: 1, 2, 3 & 6.

1 + 2 + 3 = 6.

6 is a perfect Number.

**Question 2: Show that 28 is a perfect number?**

Solution:

The factors of 28: 1, 2, 4, 7, 14 & 28.

1 + 2 + 4 + 7 + 14 = 28.

28 is a perfect Number.

**Question 3: 496 is a Perfect number or not?**

Solution:

The factors of 496: 1, 2, 4, 8, 16, 31, 62, 124, 248 & 496.

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

496 is a perfect Number.

**Question 4: Prove that 8128 is a Perfect number.**

Solution:

The factors of 8128: 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 & 8128.

The Perfect numbers can be obtained as follows:

6 = addition of the factors 1, 2 & 3.

28 = addition of the factors 1, 2, 4, 7 & 14.

496 = addition of the factors 1, 2, 4, 8, 16, 31, 62, 124 & 248.

8, 128 = addition of the factors 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032 & 4064.

33,550,336 = addition of the factors 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8191, 16382, 32764, 65528, 131056,

262112, 524224, 1048448, 2096896, 4193792, 8387584 & 16775168.