Perfect Numbers

A perfect number is a positive integer that is equal to the sum of its positive divisors, excluding itself. For instance, 28 is a perfect number because the sum of its divisors (1, 2, 4, 7, and 14) is 28.

• Perfect numbers are also known as "Complete Numbers" in number theory.
• Some of the first perfect numbers are 6, 28, 496, and 8128,
• As of 2025, a total of 52 perfect numbers have been discovered.

What are 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. A few of the starting Perfect Numbers are 6, 28, 496, 8128, and so on.

Perfect Number Examples

Perfect Numbers range from the smallest, which is 6 up to infinity, few of the starting Perfect Numbers are

• 6
• 28
• 496
• 8128
• 33550336 up to infinity.

The latest Perfect Number was discovered in 2024 and has 82,048,64 digits.

Mersenne Prime Numbers

In mathematics, a Mersenne prime is a prime number that is one less than a power of two.

It's represented as Mₙ = 2ⁿ − 1 for an integer n.

For instance, 31 is a Mersenne prime because it's 2⁵ − 1.

The initial Mersenne primes include 3, 7, 31, and 127. 45^th known Mersenne prime, discovered in 2008, is (2^37156667 − 1). Mersenne primes and perfect numbers are closely linked types of natural numbers in number theory.

Perfect Number Table

The table added below contains the starting 9 Mersenne Primes and their respective Perfect Numbers.

History of Perfect Numbers

The history of perfect number is very old it goes back to Egytian civilization as they are one who first thought about Perfect Numbers. The major development in the study of perfect numbers is credited to the Greeks, who eagerly read about Perfect numbers.

How to Find Perfect Numbers?

For example, let's consider the number 6. Its divisors are 1, 2, and 3 (excluding 6). Adding these divisors gives 1 + 2 + 3 = 6. Therefore, by definition, 6 is a perfect number.

Another example is the number 496. Its divisors (factors) are 1, 2, 4, 8, 16, 31, 62, 124, and 248 (excluding 496). Adding these divisors results in 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496, which, according to the definition, is a perfect number.

Euclid's Perfect Number Theorem

Euclid–Euler Theorem, also known as Euclid's Perfect Number Theorem, connects Perfect Numbers to Mersenne Primes. It states that an even number is perfect if and only if it can be expressed in the form [2^(p−1)(2^p − 1)] where 2^p-1 is a prime number.

Jacques Lefèvre, in 1496, suggested that the Euclid-Euler theorem encompasses all Perfect Numbers, implying the non-existence of odd Perfect Numbers.

According to Euclid's Perfect Number theorem:

2^p-1(2^p-1) is an even perfect number where we have 2^p-1 as a prime.

Similarly, we can generate the first four Perfect Numbers using the above formula (p is a prime number):

• p = 2: 2¹(2²-1) = 2 × 3 = 6
• p = 3: 2²(2³-1) = 4 × 7 = 28
• p = 5: 2⁴(2⁵-1) = 16 × 31 = 496
• p = 7: 2⁶(2⁷-1) = 64 × 127 = 8128

Perfect Number List

A few of the Perfect Numbers are 6, 28, 496, 8128, 33550336, 8589869056 the list goes on. Perfect numbers become less frequent as you go to higher values they have been studied in mathematics for centuries, and while an infinite number of them is not known, their properties and characteristics are well known.

List of All 52 Perfect Numbers

Below is a list of all the 52 perfect numbers in ascending order:

These numbers follow the pattern [2^p-1(2^p -1)] where 2^p−1 is a prime number.

Solved Questions on Perfect Numbers

Question 1: Is 28 a Perfect Number or not?

Solution:

Divisors of 28 are : 1, 2, 4, 7, 14 (excluding 28)

On adding the divisors,
1 + 2 + 4 + 7 + 14 =28

which hence proves that 28 is a Perfect Number.

Question 2: Is 56 a Perfect Number or not?

Solution:

Divisors of 56 are: 1, 2, 4, 7, 8, 14, 28 (excluding 56)

On adding the divisor,
1 + 2 + 4 + 7 + 8 + 14 + 28 = 64

which hence proves that 56 is not a Perfect Number.

Question 3: Is 8128 a Perfect Number or not?

Solution:

Divisors of 8128: 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 (excluding 8128)

On adding the divisors,
1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128

which hence proves that 8128 is a Perfect Number.

Practice Problems on Perfect Number

Question 1. The sum of all of the reciprocals of a perfect number's factors (including the perfect number itself) equals?

Question 2. The first 4 perfect numbers can be generated by using the formula 2^n-1 x [2ⁿ -1], where n = 2, 3, 5. This formula was discovered by?

Question 3. Notice the following pattern and answer: The first perfect number has one digit; the second perfect number has two digits; the third one has three digits; and the fourth one has four digits. So, does the fifth perfect number contain five digits?