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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?

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  • $\begingroup$ Did you read the Wikipedia article? "The introductory words mention (a period comprehended by) 'a perfect number' which is taken to be a reference to Plato's perfect year mentioned in his Timaeus (39d)". $\endgroup$
    – Conifold
    Dec 31, 2021 at 7:48
  • $\begingroup$ I’m voting to close this question because it was answered in the comments. $\endgroup$ Dec 31, 2021 at 12:39

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Plato does not define perfect numbers anywhere. The earliest extant definition is in Euclid VII, def. 22. Although Plato uses the term "perfect number" I do not think it certain that he understood this phrase in the same way as Euclid.

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    $\begingroup$ Or, using perfect numbers for some numerological thing that has little to do with mathematics... $\endgroup$
    – Spencer
    Dec 31, 2021 at 1:07

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