# What is the smallest perfect number

__Perfect numbers:__

**What's a perfect number?**

- A number is perfect if the sum of its divisors results in the number * 2. (That is, the divisional sum * 2 results in the number.)

**b.) Theorem of Euclid:**

- If 2
^{n}-1 is a prime number, then 2^{(n-1)}*(2^{n}-1) perfect.

**c.) Euler's theorem:**

- If a is a perfect even number, then there is an n with:
- a = 2
^{(n-1)}*(2^{n}-1). - 2
^{n}-1 is a prime number.

- a = 2

**Odd perfect numbers:**

- It is still unknown whether there are odd perfect numbers!

**Proof:**

- ts (a) = ts (2
^{(n-1)}*2^{n}-1)

= ts (2^{n-1}* ts (2^{n}-1)

=(2^{n}-1)/(2-1)*((2^{n}-1)+1)

=(2^{n}-1)*2^{n}

=(2^{n}-1)*(2*2^{(n-1)})

=2*(2^{n}-1)*(2^{-1})

= 2 * n

The first six perfect numbers: 6, 28, 496, 8,128, 33,550,336, 8,589,869,056.

- 6 has the divisors: 1, 2, 3, 6. The divisional sum is 12 because 1 + 2 + 3 + 6 = 12. That is why the number 6 is perfect because 12 = (6 * 2).

6 has the divisors: 1, 2, 3, 6. The divisional sum is 12 because 1 + 2 + 3 + 6 = 12. That is why the number 6 is perfect because 12 = (6 * 2).

28 has the divisors: 1, 2, 4, 7, 14, 28. The divisional sum is 56 because 1 + 2 + 4 + 7 + 14 + 28 = 56. That is why the number 28 is perfect because 56 = (28 * 2).

496 has the partial sum 992. Therefore the number 496 is perfect, because 992 = (496 * 2).

8,128 has the partial sum 1.6256. That is why the number 8,128 is perfect because 16,256 = (8,128 * 2).

33,550,336 has the partial sum 67,100,672. That is why the number 33,550,336 is perfect because 67,100,672 = (33,550,336 * 2).

8,589,869,056 has the partial sum 17,179,738,112. That is why the number 8,589,869,056 is perfect because 17,179,738,112 = (8,589,869,056 * 2).

6 = 2*3

28 = 2^{2}*7

496 = 2^{4}*31

8.128 = 2^{6}*127

33.550.336 = 2^{12}*8.191

8.589.869.056 = 2^{16}*131.071

*© DBG Wiehl, November 16, 1998*- How can I buy hosting from SiteGround
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