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Perfect numbers are positive integers who equal the sum of their proper integers (their aliquot sum). An well-known example is 6. Indeed, the proper divisors are 1, 2 and 3, and they sum up to 6. Another example is 28: 1+2+4+7+14.

  • 6
  • 28
  • 496
  • 8128
  • 33550336
  • 8589869056
  • 137438691328
  • … (?)

Only the first four perfect numbers were known to the ancient Greek mathematicians. Euclid could prove the following generating formula:

Two millennia later Euler proved that all even perfect numbers are of the form described by Euclid’s recipe. This means that every Mersenne prime generates an even perfect number and vice-versa.

However, what about odd perfect numbers; do they exist? No one knows. All of the 48 perfect numbers known today (see this list) happen to be even. It’s unknown as well whether there exist infinitely many perfect numbers.

Wife or mistress?

  • A poet, a priest, and a mathematician are discussing whether it's better to have a wife or a mistress.

  • Poet:

    "A mistress, because love should be free and spontaneous."

  • Priest:

    "A wife, because love should be sanctified by God."

  • Mathematician:

    "I think it's better to have both. That way, when each of them thinks you're with the other, you can do some mathematics."

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