Euclid–Euler theorem

The Euclid–Euler theorem, also known as Euler's theorem in the context of number theory, relates to the area of geometry and can be specifically described in two ways.

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n > 0

is a perfect number if and only if

n = 2^{p - 1} (2^{p} - 1) with 2^{p} - 1 prime .

p

to be a prime number, as shown under Mersenne number. It does not say that every prime exponent works, nor classify possible odd integers.^[ref]

For the forward direction, put

q = 2^{p} - 1

and use coprimality of

q

and

2^{p - 1}

σ (2^{p - 1} q) = σ (2^{p - 1}) σ (q) = (2^{p} - 1) (q + 1) = q 2^{p} = 2 n .

σ (q) = q + 1

; replacing it with a composite number changes this factor.

For a short converse outline, write

n = 2^{a} m

with

a \geq 1

and

m

an odd integer. The condition

σ (n) = 2 n

gives

σ (m) = m + t, t = \frac{m}{2 ^{a + 1} - 1} .

Here

t

m

. If

t > 1

, the distinct divisors

1, t, m

alone exceed the required sum

m + t

. Thus

t = 1

and

σ (m) = m + 1

m

to be prime and

m = 2^{a + 1} - 1