Thread: (easy?) Proof: divisors and primality

Prove:

For all pos. integers , if sum of divisors of is , then is prime.

Apparently this is famous. This is an intro course though and the proofs I found tend to be complicated. Prove it by contrapositive? Or contradiction?

Anyone know how to prove it?

The proof I found is: the sum of divisors is at least since and divide . If it's , we then know there are no other divisors. Thus, is prime.

How do we know there are no other divisors?

Originally Posted by DiscreteW

Prove:

For all pos. integers , if sum of divisors of is , then is prime.

Apparently this is famous. This is an intro course though and the proofs I found tend to be complicated. Prove it by contrapositive? Or contradiction?

Anyone know how to prove it?

The proof I found is: the sum of divisors is at least since and divide . If it's , we then know there are no other divisors. Thus, is prime.

How do we know there are no other divisors?

if there were any other divisors, the sum would be greater than (n + 1) since we would be adding more positive numbers to that number. it would have to mean that 1 and n are not the only divisors, so n would be composite if the sum of the divisors exceed (n + 1)