Write down all of its factors.

Find the sum of all the factors except for the number itself.

eg 12 has the factors 1, 2, 3, 4, 6, and 12,

so the sum of all the factors not including 12 is 16.

Does the factor-sum equal your starting number?

Are there any numbers between 1 and 30 for which this holds true?

If a number equals the sum of its factors (not counting itself),

then the number is called a perfect number.

Show that 496 is perfect.

A number than can be written as 2ⁿ - 1, where n is a natural number,

is called a Mersenne number.

What are the first ten Mersenne numbers ?

Which of these are prime?

Mersenne (who was a French mathematician in the seventeenth century) said that 2ⁿ - 1 would be prime for

n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, 257,

and for no other number less than 257.

He wasn't quite right in saying this, but close!

Some of these numbers are huge, and it was very hard to check them until recently with the help of computers.

In 1876, a mathematician called Lucas proved that 2¹²⁷ - 1 was prime,
and this remained the highest known prime for seventy years.

Show that if M = 2ⁿ - 1 is prime then

the M^th triangle number (that is M (M + 1)/2 )will be perfect.

It has been proved that all even perfect numbers are of this type.

Are there any odd perfect numbers?

If you can find one, you will be very famous!