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.

Other Examples

• 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.

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

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.