Perfect numbers

What is a perfect number and how can we calculate a perfect number?

1. MATHEMATICS
BEYOND
NOTEBOOKS
TOPIC:
PERFECT
NUMBERS…

2. 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.

3. 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.

4. 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.

5. By now, we have learned, how to check if a given
number is perfect or not.
But
How much time will you take If I ask you to list
first 10 perfect numbers?
Probably half an hour
or more…

6. Or may be much more…
Lets learn a way of finding perfect numbers
at a go…

7. Let p be a prime number
Then consider
(2
p
-1) and 2
p-1
Then,
A number k= (2
p
-1) X 2
p-1
Is always a perfect number.

8. Let us list a few prime numbers…
2,3,5,7,11,13,… etc
For p=2,
(2
p
-1) = (2
2
-1)and 2
p-1
= 2
2-1
= 4-1 = 2
1
= 3 = 2
And k =(2
p
-1) X 2
p-1
= 3 X 2
= 6 (which is a perfect number)
Note: 6 is the smallest perfect number.

9. For p=3,
(2
p
-1) = (2
3
-1) and 2
p-1
= 2
3-1
= 8-1 = 2
2
= 7 = 4
And k = (2
p
-1) X 2
p-1
= 7 X 4
= 28 (which is a perfect number)
Note: 28 is second perfect number.

10. For p=5,
(2
p
-1) = (2
5
-1) and 2
p-1
= 2
5-1
= 32-1 = 2
4
= 31 = 16
And k = (2
p
-1) X 2
p-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.

11. Now, you try to find out
next perfect number.
Then try to find first 10
perfect numbers…
Share the trick with your
friends…

12. Prepared by:
RICHA BHARDWAJ