# Perfect numbers

**Perfect numbers** are equal to the sum of their divisors that are different from the numbers themselves, for example 1 + 2 + 4 + 7 + 14 = 28. The smallest perfect number: 6 = 1 + 2 + 3. Perfect numbers grow amazingly, say, 6th Is already measured by billions (8589869056). For today, the largest, 31st perfect number (2^{216091} - 1) * 2^{216090} is known. It was obtained thanks to the discovery in September 1985 by the mathematician Marsen of the number (2^{216091} - 1), which is now known as the 2nd largest prime number. For many generations of mathematicians, perfect numbers do not give rest. For example, all the numbers found so far are even; whether the sequence of such numbers is infinite is not known. And absolutely unapproachable was the problem of perfect odd numbers.

Odd perfect numbers have not yet been detected, but it has not been proven that they do not exist. It is also unknown whether the set of all perfect numbers is infinite. It is proved that an odd perfect number, if it exists, has at least 9 distinct prime divisors and at least 75 prime divisors with multiplicity taken into account.

All even perfect numbers (except 6) are the sum of cubes of consecutive odd positive integers: *1 ^{3} + 3^{3} + 5^{3} + ... *. The sum of all the numbers that are the inverse of the divisors of a perfect number (including itself) is 2. This is a direct consequence of the definition and the fact that the sum of divisors divided by the number itself gives the sum of the numbers that are the reciprocals of the divisors. All even perfect numbers except 6 and 496 end in a decimal notation on 16, 28, 36, 56 or 76. The remainder of dividing an even perfect number different from 6 by 9 is 1.

The special character of the numbers 6 and 28 was recognized in cultures based on Abrahamic religions - claiming that God created the world in 6 days and paid attention to the fact that the Moon circulates around the Earth in about 28 days.