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 Number using the above formula (p is 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

A perfect number is a positive integer equal to the total of its positive divisors, except the number itself in number theory. For example, 6 is a perfect number since 1 + 2 + 3 equals 6.
Some of the first perfect numbers are 6, 28, 496, and 8128. Perfect numbers are also known as “Complete Numbers” and “Proper Numbers“.

This article explores perfect numbers, covering their definition, examples, Euclid’s Perfect Number theorem, and methods to find them. We’ll also address FAQs and solve examples.