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What are perfect numbers in Euclid’s number theory?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

In Euclid’s number theory, a perfect number is defined as a positive integer that is equal to the sum of its proper divisors (often referred to by Euclid as its "parts").123

In his work Elements, Euclid provided a foundational "recipe" for generating these numbers. He proved that if a series of numbers in double proportion (1, 2, 4, 8, ...) sums to a prime number, then the product of that sum and the last number in the series is a perfect number.124

Mathematically, this is expressed as:
$N = 2^{k - 1} (2^{k} - 1)$
where $2^{k} - 1$ is a prime number.3

While Euclid’s formula successfully identifies even perfect numbers, it was not until the 18th century that Leonhard Euler proved that every even perfect number must take this specific form. Whether any odd perfect numbers exist remains one of the great unsolved questions in mathematics.13