Equivalence of Definitions of Perfect Number

Proof
Consider a strictly positive integer $n$.

By Definition 1, $n$ is a perfect number $n$ equals the sum of its aliquot parts.

By definition of divisor sum function, $\map {\sigma_1} n$ equals the sum of all the divisors of $n$.

Thus $\map {\sigma_1} n - n$ equals the aliquot sum of $n$.

So by Definition 2, $n$ is a perfect number $\map {\sigma_1} n - n = n$.

Hence the definitions are equivalent.

The equivalence of definition 4 to definition 2 follows directly.

The equivalence of definition 3 to definition 2 follows from the definition of abundance.