Perfect Numbers

Perfect numbers

Any positive integer that is equal to the sum of its distinct proper factors (factors other than the number itself).

Example: 6 (proper factors: 1,2,3) is a Perfect number because 1+2+3=6.

Example: 28 (proper factors: 1,2,4,7,14) is also a Perfect number, because 1+2+4+7+14=28.

Euclid proved that 2^n-1(2ⁿ-1) is an even perfect number when 2ⁿ-1 is a Mersenne prime. These are now called Euclid numbers and Euler proved that all even Perfect numbers are of this form for some positive prime number n. Thus, 6, 28, 496 are Perfect and correspond to values of 3, 7, and 31 for 2ⁿ-1 in the formula.

This table shows the results for n=1 to 13 which include the first five Perfect numbers:

n	2n-1	2n-1(2n-1)	Perfect?	Comment
1	1	1	No	n is not prime
2	3	6	Yes	n is prime, 2n-1 is prime
3	7	28	Yes	n is prime, 2n-1 is prime
4	15	120	No	n is not prime
5	31	496	Yes	n is prime, 2n-1 is prime
6	63	2016	No	n is not prime
7	127	8128	Yes	n is prime, 2n-1 is prime
8 to 10	...	...	No	not prime
11	2047	2096128	No	n is prime, but 2n-1 is not prime
12	4095	8386560	No	n is not prime
13	8191	33550336	Yes	n is prime, 2n-1 is prime

Whether there are infinitely many even Perfect numbers or any odd perfect numbers remain unsolved questions.