Prove that, isperfect numberif isprime number

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- May 25th 2011, 07:20 AMseitPerfect number
Prove that, is

*perfect number*if is*prime number* - May 25th 2011, 07:32 AMabhishekkgp
- May 25th 2011, 07:35 AMTheEmptySet
A number is perfect if the sum of it's divisors are twice the number.

Since by assumption

is prime the has only two divisors 1 and itself.

Since the other factor is just powers of 2 its divisors are

Since the latter is a geometric series we get that

Since the has already added the divisor 1 we just add other divisor

- May 30th 2011, 07:58 AMAlso sprach Zarathustra
Also, you should know that if is an even perfect number say then and it is a prime number.

Proof:

odd, .

, hence:

(1)

is perfect number, therefor:

With (1), we get:

(2)

From the above we can deduce that , now if we put this to (2) we will get:

Or:

.

and are both divisors of m, and therefor:

.

The conclusion is that: .

From the above follows that is prime number and

Or: