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.

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.

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.

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 2^n-1(2ⁿ-1) is a perfect number. This condition can be understood using the following table. Euclid said that (2ⁿ - 1) multiplied by 2^{n - 1}, can be a perfect number if the term in the bracket, that is, (2ⁿ - 1) is a prime number. In other words, [2^{n - 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.

• 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 2^{n - 1}, we get 2^{1 - 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, 2^{n - 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, 2^{n - 1}, and (2ⁿ - 1) is a perfect number if the (2ⁿ - 1) is a prime number.

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.

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?

What are Perfect Numbers?

A perfect number is a positive integer that is equal to the sum of its factors excluding the number itself. For example, 6 is a perfect number because when we add all its factors except 6, we get, 1 + 2 + 3 = 6. We get the sum as the number itself. Therefore, 6 is a perfect number.

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. However, till now, we know only 51 perfect numbers.

What are the First 10 Perfect Numbers?

According to Euclid's proposition, if 2ⁿ -1 is a prime number, then 2^n-1(2ⁿ-1) is a perfect number. The first 10 perfect numbers are:

What is the Largest Perfect Number?

The largest perfect number is unknown.

Is 9 a Perfect Number?

No, 9 is not a perfect number because its factors (1 and 3) excluding 9 do not sum up to 9. They sum up to 4. And we know that the sum of the factors of Perfect numbers is always the number itself.

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?

There are around 51 known perfect numbers. The latest perfect number was discovered in 2018 which has 49,724,095 digits.

What are the Perfect Numbers Between 1 to 100?

There are two perfect numbers between 1 to 100. They are 6 and 28.