We call a positive integer perfect if it equals the sum of its positive divisors other than itself. (a) Prove that 6 and 28 are perfect numbers (b) Prove that if 2p − 1 is prime, then 2p−1
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The divisors of are . If we sum these divisors then we have ,
, Therefore, 6 is a perfect number.
The divisors of are . Summing these divisors gives,
. This shows that is a perfect number.
We need to show that If is a prime number, then is a perfect number.
Let be an integer so that is a prime number. Also, let so that . We can write the divisors of as, . Since
is prime, we can write out the other divisors of as, .
Let us recall that,
, where so,
We can use the same formula we have used to find for,
We can see that,
We can now sum all the divisors of by combining equations and ,
Therefore, if is prime then is perfect.