What are perfect numbers?

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About 500 years Before Christ, Pythagoras stood out as a great mathematician who revealed great mysteries and he reached incredible mathematical conclusions that are still used nowadays, among which one of the most famous is the “Pythagoras’s theorem”. The Pythagoras’s disciples are known as Pythagoreans and they were also known because they liked riddles and mathematical enigmas for lots of which the solution has not been found yet.


The Pythagoreans were the ones who called Perfect Number to every natural number which is equal to the sum of all its divisors, being itself excluded from this set of divisors, the number itself. Let´s have a look at some examples:

  • 6 = 1 + 2 + 3
  • 28 = 1 + 2 + 4 + 7 + 14
  • 496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
  • 8128
  • 33550336
  • 8589869056
  • ...
  • 2658455991569831744654692615953842176

The first four perfect numbers (6, 28, 496 and 8128) were the only ones that the ancient Greek people knew since at least Euclid’s. In the XV century the number 33,550,336 was added to the list. As we can notice through the example above, the perfect numbers are extremely rare. Only after the invention of the computer, it was possible to find bigger perfect numbers. The thirtieth known perfect number is 2,658,455,991,569,831,744,645,692,615,953,842,176 which has 37 figures. And the forty-fourth perfect number that was found has almost 20 million figures! Isn’t it amazing?

The existence or non-existence of odd perfect numbers remains a challenge for the Theory of numbers. And if there is one or none, it is an old picture that remains without a solution.

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