What Are Perfect Numbers? Definition, Examples, And Facts

Perfect Numbers

In number theory, a perfect number is a positive integer that is equal to the sum of its positive factors, excluding the number itself. The most popular and the smallest perfect number is 6, which is equal to the sum of 1, 2, and 3. Other examples of perfect numbers are 28, 496, and 8128.

1.	What are Perfect Numbers?
2.	History of Perfect Numbers
3.	How to Find a Perfect Number?
4.	Perfect Numbers List in a Table
5.	FAQs on Perfect Numbers

What are Perfect Numbers?

A perfect number is a positive integer that is equal to the sum of its factors except for the number itself. In other words, perfect numbers are the positive integers that are the sum of its divisors. The smallest perfect number is 6, which is the sum of its factors: 1, 2, and 3. It is to be noted that this sum does not include the number itself which is also a factor of itself.

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.

History of Perfect Numbers

The initial study of perfect numbers may go back to the Egyptians who might have come across such numbers naturally. 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. This number 6 gathered much attention in the beginning by the Pythagoreans, more for its mystical and numerological properties than for any mathematical significance. It is to be noted that 6 is the smallest perfect number, the next being 28.

How to Find a Perfect Number?

In order to find a perfect number, we can use the technique told by Euclid. According to Euclid, there is an expression that can be a perfect number subject to a specific condition. According to his proposition, if 2n -1 is a prime number, then 2n-1(2n-1) is a perfect number. This condition can be understood using the following table. Euclid said that (2n - 1) multiplied by 2n - 1, can be a perfect number if the term in the bracket, that is, (2n - 1) is a prime number. In other words, [2n - 1 × (2n - 1) = perfect number], if (2n - 1) is a prime number.

Therefore, we need to find a value of 'n' for which (2n - 1) is prime. So, the following table will help us understand this better. Let us follow the steps given below so that we can relate to the table and understand the process.

n	2n - 1	(2n - 1)	2n - 1 × (2n - 1)
1	1	1	-
2	2	3 (prime number)	6 (perfect number)
3	4	7 (prime number)	28 (perfect number)
4	8	15	-
5	16	31 (prime number)	496 (perfect number)
6	32	63	-
7	64	127 (prime number)	8128 (perfect number)
8	128	255	-
9	256	511	-
10	512	1023	-

• Step 1: Let us start with n = 1. After substituting the value of n = 1 in both the expressions, we will see the results. If we substitute 1 in 2n - 1, we get 21 - 1 = 20 = 1. And substituting 1 in (2n - 1), we get, (21 - 1) = 1.
• Step 2: After substituting n = 2, n = 3, n = 4, and so on we get the resultant numbers written in the table.
• Step 3: Now, we need to observe the column of (2n - 1), in which if the number is a prime number, then the product of the two expressions, 2n - 1 and (2n - 1) will result in a perfect number.
• Step 4: For example, if we take n = 2, we get 2 as the result of the first expression, and we get 3 in the second expression. When we take n = 3, we get 4 as the result of the first expression, and 7 in the second expression.
• Step 5: After listing out the numbers as given, we need to observe those numbers under the column (2n - 1) that are prime numbers. And the respective products of these will always result in a perfect number. In the table, we can see that 3, 7, 31, 127 are prime numbers, this means that their respective products shown in the next column will always be perfect numbers, that is, 6, 28, 496, and 8128 are perfect numbers. This means the product of the two factors, 2n - 1, and (2n - 1) is a perfect number if the (2n - 1) is a prime number.

Perfect Numbers List in a Table

A few perfect numbers 6, 28, 496 and 8128 are known to us since ancient times. Let us see their divisors and their sum through the table given below. Their sum results in the number itself. Therefore, these are known as perfect numbers.

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

Important Notes

• Perfect numbers are the positive integers which are the sum of their proper divisors.
• The smallest perfect number is 6.
• All the perfect numbers are even numbers. It is still unknown whether odd perfect numbers exist or not.
• All the perfect numbers end in 6 and 8 alternatively.

☛ Related Topics

• Perfect Number Calculator
• Prime Number Calculator
• How to Know if a Number is a Perfect Square?