MATHEMATICS BEYOND

NOTEBOOKS

TOPIC: PERFECT NUMBERS…

DEFINITION

If sum of all the factors of a number (other than the number itself) comes out to

be the number itself, then the number is said to be a perfect number.

For example

Factors of 6= 1,2,3 and 6

We observe that 6=1+2+3 (6, the number

itself is excluded)

Thus 6 is a PERFECT NUMBER.

In fact 6 is the smallest perfect number.

Similarly ,

Factors of 28= 1, 2, 4, 7, 14 and 28

And 28 = 1+2+4+7+14 (28 excluded)

Thus 28 is a perfect number.

By now, we have learned, how to check if a given number is perfect or not.

ButHow much time will you take If I ask you to list

first 10 perfect numbers?

Probably half an hour or more…

Or may be much more…

Lets learn a way of finding perfect numbers at a go…

Let p be a prime number

Then consider(2p-1) and 2p-1

Then, A number k= (2p-1) X 2p-1

Is always a perfect number.

Let us list a few prime numbers…2,3,5,7,11,13,… etc

For p=2,

(2p-1) = (22-1) and 2p-1 = 22-1

= 4-1 = 21

= 3 = 2

And k = (2p-1) X 2p-1

= 3 X 2= 6 (which is a perfect number)

Note: 6 is the smallest perfect number.

For p=3,

(2p-1) = (23-1) and 2p-1 = 23-1

= 8-1 = 22

= 7 = 4

And k = (2p-1) X 2p-1

= 7 X 4= 28 (which is a perfect number)

Note: 28 is second perfect number.

For p=5,

(2p-1) = (25-1) and 2p-1 = 25-1

= 32-1 = 24

= 31 = 16

And k = (2p-1) X 2p-1

= 31 X 16= 496 (which is a perfect number)

Note: 496 is third perfect number and there does not exists any perfect number between 28 and 496.

Now, you try to find out next perfect number.

Then try to find first 10 perfect numbers…

Share the trick with your friends…

Prepared by:RICHA BHARDWAJ