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 . . .
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* . The n's have no pattern.
Hello, ecMathGeek!
A number is perfect if the sum of its proper divisors equals the number.I have 2 follow-up questions:
What is a "perfect number"? By this I mean: What makes it "perfect"?
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.
If there is no pattern, they cannot be "derived".How are the n's derived if there's no pattern to them
(other than that they seem to be prime numbers)?
. . It's simply trial-and-error.
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.