What are Perfect Numbers?

“Nobody is perfect,” they say. Yes, nothing might be perfect in the real applied world. But in the field of mathematics and theoretical assumptions, thousands of things are perfect. One of such things is a perfect number. So what is a perfect number? What makes it perfect? How perfect it is?

According to Wikipedia, a perfect number is:

  1. A positive integer that is equal to the sum of its proper positive divisors excluding the number itself, also known as its aliquot sum; or
  2. A positive number that is half the sum of all of its positive divisors including itself.

The smallest perfect number is 6. How does it become a perfect number?

Let us first find out the POSITIVE divisors of number 6. But what are divisors? These are numbers which when multiplied will result to number 6. The numbers 2 and 3 are divisors of number 6 since their product is equal to 6, as well as the numbers 1 and 6. Hence, the divisors for number 6 collectively are 1, 2, 3 and 6.

From the first definition, a perfect number is one whose total of divisors other than itself is equal to it. For number 6, its divisors other than itself are 1, 2 and 3. Adding up the three numbers (1+2+3) will result to 6. And that is a perfect number!

Equivalently, the second definition says that a perfect number is half the sum of its divisors. For number 6, the total of all its divisors (1+2+3+6) is equal to 12. Half of that is 6, equal to the number itself. And that again is a perfect number!

That is how a number becomes perfect. Can you guess now which perfect number is next to 6? Well, it is number 28!

The positive divisors of 28 are 1,2,4,7,14, and 28. Adding up all the digits result to 56 and taking half of that is equal to the number itself – 28. The third and fourth numbers are quite very far from them, they are numbers 496 and 8128.

The idea of perfect numbers went back as early as 100 AD to the early Greek mathematics and the mathematician Nicomachus who discovered the first four perfect numbers. Only in 1456 did the fifth perfect number, which is 33,550,336, is identified. In 1588, the Italian mathematician Pietro Cataldi noted the sixth and 7th perfect number, respectively 8,589,869,056 and 137,438,691,328.

Wow, did it take really that long to discover these perfect numbers? I suppose perfection is such a rare beauty!

However, new studies revealed the equations for solving these perfect numbers. The perfect numbers are derived by the formula

2p-1 (2p – 1)

where p is a prime number. For example, the first four perfect numbers are generated as follows:

for p = 2:   21(22−1) = 6
for p = 3:   22(23−1) = 28
for p = 5:   24(25−1) = 496
for p = 7:   26(27−1) = 8128.

The first 40 prime numbers can be applied to the equation.

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