They are those natural numbers that have the property of being equal to the sum of their own divisors. So, for example, the number 6 (1 + 2 + 3 = 6) is a perfect number. The even perfect numbers are given by the following expression, known as Euclid-Euler theorem:
n = 2n-1(2n - 1)
In which term 2n - 1 must be a prime number. By making n = 2,3,5,7,13,17 in the previous formula, we obtain the first six even perfect numbers: 6; 28; 496; 8128; 33550336; 8589869056. To date 49 even perfect numbers are known, no odd numbers have been found, although that does not mean that they do not exist.
A perfect number is that natural number that is equal to the sum of its positive proper divisors. For example, 6 is a perfect number, since when considering its proper divisors (1,2,3), we obtain:
1 + 2 + 3 = 6
Let A and B be two positive integers; both will be friendly numbers if the sum of the proper divisors of one equals the other and vice versa. For example, 220 and 284:
- The divisors of 220 are: 1,2,4,5,10,11,20,22,44,55 and 110, which together add up to 284.
- The proper divisors of 284 are: 1,2,4,71,142 and 110, which together add up to 220.
Now if we relate it to friendly numbers, a perfect number is friendly to itself, since it is equal to the sum of its proper divisors.
How are the perfect numbers calculated?
To find the perfect numbers, the Euclid - Euler theorem is used:
If n is a perfect and even number, then
n = 2n-1(2n - 1)
Where term 2n - 1 is always a prime number. According to the above, the smallest perfect number is 6:
1 + 2 + 3=6
The following is 28:
1 + 2 + 4 + 7 + 14 = 28
After this one, no other appears until 496, the fourth perfect number is 8128, the fifth perfect is 33550336. Until January 2016, 49 perfect numbers are known, the last one corresponds to n = 74207281 with which it obtains a number with 44677235 digits!
Is there an odd number?
So far all the perfect numbers found are even, but that doesn't mean they don't exist.