Proposition IX.36 — If as many numbers as we please beginning from a unit be set…

Proof

Let

s = 1 + 2 + 4 + \dots + 2^{n - 1} = 2^{n} - 1

be prime (a Mersenne prime); set

N = s \cdot 2^{n - 1}

. The proper divisors of

N

are

1, 2, 4, \dots, 2^{n - 1}, s, 2 s, \dots, 2^{n - 2} s

, whose sum is

(2^{n} - 1) + s (2^{n - 1} - 1) = s + s (2^{n - 1} - 1) = s \cdot 2^{n - 1} = N

. So

N

equals the sum of its proper divisors and is perfect.

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Depends on (4)

IX.35Proposition IX.35If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last…
VII.34Proposition VII.34Given two numbers, to find the least number which they measure.
VII.11Definition VII.11A prime number is that which is measured by a unit alone.
VII.22Definition VII.22A perfect number is that which is equal to the sum of its own parts (its proper divisors).

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