Perfect Numbers
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. The smallest perfect number is 6, which is equal to the sum of 1,2, and 3. Other perfect numbers are 28, 496, and 8128. It is not known when perfect numbers were first studied and indeed the first studies may go back to the earliest times when numbers first aroused curiosity. It is quite likely, although not certain, that the Egyptians would have come across such numbers naturally given the way their methods of calculation worked
History of Perfect Numbers
There is not much information regarding the discovery of perfect numbers. It is said that perhaps the Egyptians may have discovered them. Despite knowing the existence of perfect numbers, it was only the Greeks who were eager to study more about these numbers. Perfect numbers were studied by Pythagoras and his followers for its mystical properties. The smallest perfect number found was 6. It is still unknown whether there are infinite perfect numbers. The number 6 gathered much attention in the beginning by the Pythagoreans, more for its mystical and numerological properties than for any mathematical significance, as it is the sum of its proper factors, i.e. 6 = 1 + 2 + 3. This is the smallest Perfect Number, the next being 28.
The Pythagoreans were most interested in the occult properties of these numbers, they did little of mathematical significance with them. It was around 300 BC, when Euclid wrote his Elements that the first real result was made. Although Euclid concentrated on Geometry, many number theory results can be found in his text (Burton, 1980).
What are Perfect Numbers?
Perfect numbers are the positive integers that are equal to the sum of its factors except for the number itself. In other words, perfect numbers are the positive integers which are the sum of its proper divisors. The smallest perfect number is 6, which is the sum of its proper divisors: 1, 2 and 3
Do you know that when the sum of all the divisors of a number is equal to twice the number, the number has a separate name? Such numbers are called complete numbers. In fact, all the perfect numbers are also complete numbers.
Perfect Numbers List in a Table
The four perfect numbers 6, 28, 496 and 8128 are known to us since ancient times. Let's see their divisors and their sum through the table given below:
Perfect Number  Sum of Divisors 

6  1 + 2 + 3 
28  1 + 2 + 4 + 7 + 14 
496  1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 
8128  1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 
How to Find a Perfect Number?
Perfect numbers are the special type of numbers that are less known to students as compared to the other types of numbers. In this section, you will learn how to find perfect numbers easily.
 There is no single fixed preposition to find perfect numbers.
 The first four perfect numbers are generated by the formula 2^{p−1}(2^{p} − 1), with p a prime number.
 Prime numbers of the form 2^{p} − 1 are known as Mersenne Primes, after the seventeenthcentury monk Marin Mersenne.
Some of them are given below.
Proposition 1
If as many numbers as those beginning from the unit be set out continuously in double proportion until the sum of all becomes a prime, then the product of the sum and the last number makes a perfect number. Double Proportion means that each number in a sequence is double the preceding number. For example, 1+2+4=7 is a prime number.
According to Proposition 1, sum × last number = perfect number
7 × 4=28
And yes! 28 is a perfect number.
Proposition 2
If N is a Mersenne prime, then \(\begin{align}\frac{N(N+1)}{2}\end{align}\) is a perfect number. Mersenne prime is a prime that is one less than the power of 2. For example, Let's take N as 31 which is 1 less than 2^{4}. Then,
\(\begin{align}\frac{N(N+1)}{2}=\frac{31(31+1)}{2}=496\end{align}\)
And 496 is a perfect number.
Topics related to Perfect Numbers
Here are a few topics that can be referred to for more information on topics related to perfect numbers.
Tips and Tricks
 The k^{th} perfect number has k digits.
 All the perfect numbers are even. It is still unknown whether odd perfect numbers exist or not.
 All the perfect numbers end in 6 and 8 alternatively.
 The sum of all the divisors of a perfect number gives the result as twice the perfect number.
Important Notes
 Perfect numbers are the positive integers which are the sum of their proper divisors.
 The smallest perfect number is 6
 If 2^{k} 1 is prime for k>1, then 2^{k1}(2^{k} 1) is a perfect number.
 The only squarefree perfect number is 6
 A perfect number is called Ore Harmonic number if the harmonic mean of its divisor is an integer.
Solved Examples

Example 1
Is 28 a perfect number?
Solution:
The proper factors of 28 are 1, 2, 4, 7 and 14. The sum of proper factors is 28. According to the definition of perfect numbers, 28 is a perfect number. therefore, 28 is a perfect number.

Example 2
According to Euclid's proposition, if 2^{p} 1 is a prime number, then 2^{p1}(2^{p}1) is a perfect number. Can you make a list of 8 perfect numbers using this proposition?
Solution:
The first 8 Mersenne Primes are 2, 3, 5, 7, 13, 17, 19 and 31.
Prime, p 2^{p}1 2^{p1} Perfect Number, 2^{p1}(2^{p}1) 2 3 2 6 3 7 4 28 5 31 16 496 7 127 64 8128 13 8191 4096 33550336 17 131071 65536 8589869056 19 524287 262144 137438691328 31 2147483647 1073741824 2305843008139952128 Therefore, the abovementioned are the first 8 perfect numbers.
FAQs on Perfect Numbers
What are the First 5 Perfect Numbers?
The first 5 perfect numbers are 6, 28, 496, 8128, and 33550336.
How Many Perfect Numbers are There?
It is still unknown whether there are infinite perfect numbers or not. But till now, we know only 51 perfect numbers.
What are the First 10 Perfect Numbers?
According to Euclid's proposition, if 2^{p} 1 is a prime number, then 2^{p1}(2^{p}1) is a perfect number. The first 10 perfect numbers are:
6 
28 
496 
8128 
33550336 
8589869056 
137438691328 
2305843008 139952128 
26584559915698317446 54692615953842176 
1915619426082361072947933780 84303638130997321548169216 
What is the Largest Perfect Number?
The largest perfect number is unknown.
Is 9 a Perfect Number?
No, 9 is not a perfect number as the factors of 9 excluding it are 1 and 3 which is not added to 9.
What is the Smallest Perfect Number?
The smallest perfect number is 6, which is the sum of 1,2,3.
How Many Perfect Numbers are there and What are the Perfect Numbers from 1 to 100?
There are around 51 known perfect numbers. There are only 2 perfect numbers from 1 to 100 which are 6 and 28. The latest perfect number was discovered in 2018 which has 49,724,095 digits.