I am new to proper mathematical proofs and was attempting to prove:
A prime number cannot be a perfect number.
Here is my go:
suppose n is a prime number.
Therefore the only divisors of n, are n and 1 by the definition of a prime number.
Now suppose n is a perfect number as well.
By the definition of a perfect number we can say n = 1, as the only positive integer divisor of n that is less than n is 1.
But there are no positive integer divisors of 1 that are less than 1. So 1 = 0 by definition of a perfect number.
Since 1 ≠ 0 we have reached the contradiction, and therefor we can say that a number cannot have both properties of being prime and perfect.
Is this proof valid? If not why? And in general is something blatantly wrong like n = m a “good enough” proof by contradiction if we already new beforehand that n ≠ m