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What is a perfect number?

A perfect number is a whole number, an integer greater than zero; and when you add up all of the factors less than that number, you get that number. For example:

The factors of 6 are 1, 2, 3 and 6.
1 + 2 + 3 = 6

The factors of 28 are 1, 2, 4, 7, 14 and 28.
1 + 2 + 4 + 7 + 14 = 28

The factors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248 and 496.
1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496

The factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064 and 8128.

The first four perfect numbers were known over 2,000 years ago. Some ancient cultures gave mystic interpretations to numbers that they thought were magic. Using a computer program, eventually ran for a billion of time the following 5 perfect numbers can be found:

6 = 1+2+3
28 = 1+2+4+7+14
496 = 1+2+4+8+16+31+62+124+248
8,128 = 1+2+4+8+16+32+64+127+254+508+1016+2032+4064

All of the perfect numbers that have been found so far fit the formula

2^(n-1) * ( 2^n - 1 )

where "n" is one of a very short list of prime numbers that can be used to create "Mersenne" prime numbers.

2^1 * ( 2^2 - 1 ) = 2 * 3 = 6
2^2 * ( 2^3 - 1 ) = 4 * 7 = 28
2^4 * ( 2^5 - 1 ) = 16 * 31 = 496
2^6 * ( 2^7 - 1 ) = 64 * 127 = 8,128
2^12 * (2^13-1) = 4096 * 8191 = 33,550,336
2^16 * (2^17-1) = 65536 * 131071 = 8,589,869,056
2^18 * (2^19-1) = 262144 * 524287 = 137,438,691,328
2^30 * (2^31-1) = 1073741824*2147483647=2,305,843,008,139,952,128

How many perfect numbers are there?

We do not know how many perfect numbers there are. We do know that there are an infinite number of prime numbers, which means there is a very high chance that there are an infinite number of perfect numbers. This is because there is a strong link between perfect numbers and a certain kind of prime number (the Mersenne primes).

How many perfect numbers are known?

So far, according to the Mersenne organization, there are 37 known Mersenne prime numbers. This means that there are 37 known "perfect" numbers. The newest prime number was discovered on January 27, 1998. Obviously, this is not all of them. There are many that we will never know. We don't even know if the 37th number found is the 37th number; there may be another perfect number between the 35th and the 36th. It is very likely, too, that there are many more that we will NEVER know.

Are there any odd perfect numbers?

Nobody has found any odd perfect numbers, but we do not know if any odd ones exist. If any odd perfect numbers exist, they are not based on the "Mersenne" method of calculating perfect numbers.



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