I've read several texts suggesting that Plato defines "perfect number" in his Republic, book VIII 546 b. However, there's no definition as we can see from - for example - this translation: "... Now that which is of divine birth has a period which is contained in a perfect number$^1$ but the period of human birth..."
"1: i.e. a cyclical number, such as 6, which is equal to the sum of its divisors 1, 2, 3, so that when the circle or time represented by 6 is completed, the lesser times or rotations represented by 1, 2, 3 are also completed."
So it's just the footnotes that explain what the perfect number is. Where does Plato himself define it?