Perfect Numbers
A perfect number is defined as a number which is equal to the sum of its positive factors.
Let’s take an example –
The first perfect number (integer) is 6.
A factor of 6 are – 1, 2, 3.
Sum of 1, 2, 3, – 1+2+3 = 6.
Therefore, 6 is a perfect number.
In order to find the perfect number, the number itself should not be included in the list of factors.
Let’s take a few more examples:-
- N=28
Factors of 28 = 1, 2, 4, 7, 14
Sum = 1+2+4+7+14 =28
- N=496
Factors of 496 =1, 2, 4, 8, 16, 31, 62, 124, 248
Sum= 1+2+4+8+16+31+62+124+248 = 496
First, five perfect numbers are –
6, 28, 496, 8128, 33550336
We know that (2^k – 1) is a form of a prime number, then 2^(k-1)(2^k – 1) is a form of the perfect number.
Where k must be a prime number.
We can write :-
6 = 2^1(2^2 – 1)
28 = 2^2(2^3 – 1)
Magic of perfect numbers can be seen in our nature also.
Moons period is of 28 days.
