Thread: Even Perfect Numbers

How do you find the first 6 smallest even perfect numbers? What are they?

Hello, Ideasman!

Who dared to ask this question? . . . and why?

How do you find the first 6 smallest even perfect numbers?
. . What are they?

Euclid's formula for even perfect numbers:

. . N .= .(2^{n-1})·(2^n - 1), .where the second factor is prime.


The first five values are: .n .= .2, 3, 5, 7, 13 .*

. . which produces: .N .= .6, 28, 496, 8128, 33550336


I'll let you find the sixth one . . .


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

* . The n's have no pattern.

Originally Posted by Soroban

Hello, Ideasman!

Who dared to ask this question? . . . and why?


Euclid's formula for even perfect numbers:

. . N .= .(2^{n-1})·(2^n - 1), .where the second factor is prime.


The first five values are: .n .= .2, 3, 5, 7, 13 .*

. . which produces: .N .= .6, 28, 496, 8128, 33550336


I'll let you find the sixth one . . .


~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

* . The n's have no pattern.

I have 2 follow-up questions:
What is a "perfect number"? By this I mean: What makes it "perfect"?
How are the n's derived if there's no pattern to them (other than that they seem to be prime numbers)?

Hello, ecMathGeek!

I have 2 follow-up questions:
What is a "perfect number"? By this I mean: What makes it "perfect"?

A number is perfect if the sum of its proper divisors equals the number.

Examples: .6 .= .1 + 2 + 3
. . - . - . . 28 .=. 1 + 2 + 4 + 7 + 14
. - . - . - 496 .= .1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248

Therefore: .6, 28, and 496 are "perfect" numbers.

How are the n's derived if there's no pattern to them
(other than that they seem to be prime numbers)?

If there is no pattern, they cannot be "derived".
. . It's simply trial-and-error.

Originally Posted by ecMathGeek

I have 2 follow-up questions:
What is a "perfect number"? By this I mean: What makes it "perfect"?
How are the n's derived if there's no pattern to them (other than that they seem to be prime numbers)?

There are many applications to perfect numbers. The prime that is used in the equation is called a Mersenne Prime. There is a nice theorem out (I can prove it if you wish) that states there is an infinite number of perfect numbers. However, one of the questions that still remains is whether all are even. No one knows.

Originally Posted by AfterShock

There are many applications to perfect numbers. The prime that is used in the equation is called a Mersenne Prime. There is a nice theorem out (I can prove it if you wish) that states there is an infinite number of perfect numbers. However, one of the questions that still remains is whether all are even. No one knows.

No, it is an unsolved problem. It is not know whether of not Mersennse primes are infinite.
Which will imply the infinitude of even perfect numbers.