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
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: