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Proposition: 9.36: Theorem of Even Perfect Numbers (First Part)

Euclid's Formulation

If any multitude whatsoever of numbers is set out continuously in a double proportion, (starting) from a unit, until the whole sum added together becomes prime, and the sum multiplied into the last (number) makes some (number), then the (number so) created will be perfect. * For let any multitude of numbers, $A$, $B$, $C$, $D$, be set out (continuouly) in a double proportion, until the whole sum added together is made prime.

• And let $E$ be equal to the sum.
• And let $E$ make $FG$ (by) multiplying $D$.
• I say that $FG$ is a perfect (number) .

Historical Notes

• The ancient Greeks knew of four perfect numbers: 6, 28, 496, and 8128, which correspond to $n= 2$, 3, 5, and 7, respectively.

Modern Formulation

See even perfect numbers.

Table of Contents

Proofs: 1

Mentioned in:

Propositions: 1

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References

Adapted from (subject to copyright, with kind permission)

1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016