Perfect numbers

A perfect number is a natural number a, whose sum of all its divisors d₁, d₂, d₃, … d_n is equal to twice the given perfect number a.

2a = d₁ + d₂ + d₃ + … + d_n

The smallest perfect number is a = 6. This number is divisible by 1, 2, 3, 6

2·6 = 1 + 2 + 3 + 6

The next perfect number is 28. It is divisible by 1, 2, 4, 7, 14, 28

2·28 = 1 + 2 + 4 + 7 + 14 + 28

Perfect numbers were also studied by the greatest ancient Greek mathematician Euclid (365 – 300 BC). In the 10th book of his *Stoicheia* (Elements) he gives a formula for calculating perfect numbers:

a = 2^n-1(2ⁿ – 1) where n and (2ⁿ – 1) are prime numbers.

n = 2 : a = 2¹(2² – 1) = 2·3 = 6

n = 3 : a = 2²(2³ – 1) = 4·7 = 28

n = 4 : a = 2³(2⁴ – 1) = 8·15 = 120 , 120 is not a perfect number because 4 and 15 are not primes

n = 5 : a = 2⁴(2⁵ – 1) = 16·31 = 496

n = 7 : a = 2⁶(2⁷ – 1) = 64·127 = 8128

At present we know 26 perfect numbers for exponents n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 44497. (These are primes.) All known perfect numbers are even.

The largest currently known perfect number is a = 2⁴⁴⁴⁹⁶(2⁴⁴⁴⁹⁷ – 1).

Literature: Opava, Z.: Mathematics Around Us, Prague, Albatros 1989

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