Learning Perfect Number

Any number which is a positive integer  is called a perfect number if the sum of the factors(or divisors) of that number is equal to that number itself. Obviously, the factors excludes that number itself. In other words, we can say that a perfect number is an integer whose sum of factors(or divisors) is double the number.


Example of a perfect number

6

Factors of 6 are 1,2,3,6

Sum of the factors of 6(excluding 6) = 1+2+3 = 6 (the number itself)

Hence 6 is a perfect number.

Similarly, 28( factors of 28 excluding itself are 1,2,4,7 and 14, 1+2+4+7+14=28) is a perfect number.

Having problem with Rational and Irrational Numbers keep reading my upcoming posts, i will try to help you.

General form of a perfect number


The general form of a perfect number is given as 2p-1 (2p - 1), where p is a prime number. It is important to note that not all numbers of that form are perfect number but all perfect numbers will be of that form.

for p =2, 2p-1 (2p - 1) = 21 (22  - 1) =2*3 = 6, a perfect number.

Perfect Number Examples

Definition:

A perfect number is the positive integer in that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself. Equivalently of a perfect number is that a number is half the sum of all of its positive divisors (including itself), or σ(n) = 2n.


Even perfect numbers

In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes, after the seventeenth-century monk Marin Mersenne, who studied number theory and perfect numbers. However, not all numbers of the form 2p − 1 with p a prime are prime

In the first four perfect numbers are generated by the formula 2p−1(2p − 1), with p a prime number:

for p = 2: 21(22 − 1) = 6

for p = 3: 22(23 − 1) = 28

for p = 5: 24(25 − 1) = 496

for p = 7: 26(27 − 1) = 8128.

Noticing that 2p − 1 is a prime number in each instance, Euclid proved that the formula 2p−1(2p − 1) gives an even perfect number In order for 2p − 1 to be prime, it is necessary that p itself is prime. Prime numbers of the form 2p − 1 are known as Mersenne primes. I have recently faced lot of problem while learning Definition of Rational Numbers, But thank to online resources of math which helped me to learn myself easily on net.


Odd perfect numbers

It is unknown whether there are any odd perfect numbers. Various of results have been obtained, but none of that has helped to locate one or otherwise resolve the question of their existence. Carl Pomerance has been presented by a heuristic argument in which suggests that no odd perfect numbers exist.[4] Also, it can been conjectured of that there are no odd Ore's harmonic numbers, except for 1. If true in this would to imply that there are no odd perfect numbers.