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Perfect Numbers

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 Sub Topics There are different types of numbers in the number system. Perfect numbers are one of such types. Perfect number is a whole number, a positive number whose factors when added give back the number as answer. The factors of the numbers are less than that number. Perfect number is a number that is half the sum of all of its positive divisors. The first four perfect numbers are 6, 28, 496, 8128. Smallest perfect number is 6, which is the sum of 1, 2 and 3. Pythagoreans coined the term 'perfect numbers' and the first four perfect numbers were known over 2,000 years ago. Perfect numbers are pretty rare. The 3 and 4 numbers look far apart, the same continues and the distance between 4 and 5 is large as well. There are infinite number of perfect numbers.

Even Perfect Numbers

According to Euclid, whenever 2$^{p}$ - 1 is prime, then it is proved that 2$^{p - 1}$(2$^{p}$ -1) is an even perfect number. It is necessary that p itself be prime for 2$^{p}$ - 1 to be prime.
Prime numbers of the form 2$^{p}$ - 1 are known as Mersenne primes, where p is a prime number. There is one to one relationship between even perfect numbers and mersenne primes, each mersenne prime generates one even perfect number and vice versa. This is known as Euclid-Euler theorem. 48 Mersenne primes and 48 even perfect numbers are known. The largest of these is 2$^{57885160}$ x (2$^{57885161-1}$). Therefore, perfect number is calculated from the mersenne prime number.

Odd Perfect Numbers

It is guessed that there are no odd perfect numbers. If there are some, then they are quite large over 300 digits and have numerous prime factors and in the form of $P_{1}^{2e_{1}}$. $P_{2}^{2e_{2}}$. ........ $P_{n}^{2e_{n}}$, where, $P_{1}$, $P_{2}$,...... $P_{n}$ are the distinct prime number. The largest prime factor of an odd number must be atleast 100000007, 10007 and 101. The smallest prime factor of an odd perfect number with all even powers less than six, determined an upper bound of exp $(4.97401 \times 10^{10}$)

Any odd perfect number N must satisfy the following conditions:
1. N > 101500.
2. N is not divisible by 105.
3. Smallest prime factor of N is less than $\frac{(2p+8)}{3}$.
4. N < 2$^{4^{p + 1}}$.
5. N has at least 101 prime factors and at least 9 distinct prime factors. If 3 is not one of the factors of N, then N has at least 12 distinct prime factors.

How to Find Perfect Numbers?

Perfect numbers can be found by using the formula given by Euclid as (2$^{p-1}$) (2$^{p}$ - 1), with 'p' being prime. Given below are the examples of perfect prime numbers.

Solved Examples

Question 1: Find the fifth perfect prime number.
Solution:
Step 1: The fifth prime number is 11(2,3,5,7,11)
The formula to find perfect prime number is given below
(2$^{p-1}$) (2$^{p}$ - 1)

Plug in 11 in the above formula, we get
(2$^{11-1}$) (2$^{11}$ - 1)
= 4096 x 8191
= 33550336

Step 2: The fifth prime number is 11(2,3,5,7,11)
The formula to find perfect prime number is given below
(2$^{p-1}$) (2$^{p}$ - 1)

Plug in 11 in the above formula, we get
(2$^{11-1}$) (2$^{11}$ - 1)
= 4096 x 8191
= 33550336

Question 2: List first 10 perfect prime numbers.
Solution:
The first ten prime numbers are 2, 3, 5, 7, 13, 17, 19, 31, 61, 89.
The formula to find perfect prime number is given below:
(2$^{p-1}$) (2$^{p}$ - 1)

Putting 2 in the above formula, we get
(2$^{2-1}$) (2$^{2}$ - 1)
= 2 x 3
= 6

Similarly, substituting the value of prime numbers in the above formula,  we get the first ten perfect prime numbers which are given below:
6, 28, 496, 8128, 33550336, 8589869056,137438691328, 2305843008139952128, 2658455991569831744654692615953842176, 191561942608236107294793378084303638130997321548169216.