A Perfect Number is a positive integer which is
equal to the sum all its positive divisors including one but excluding itself.
The smallest perfect number is 6, since
6 = 1 + 2 + 3.
and the divisors of 6 excluding itself
are 1, 2 and 3.
Equivalently, a perfect number is also
equal to half of the sum of all its positive divisors including one and itself.
For example,
6 = (1 + 2 + 3 + 6)/2
The next larger perfect number is 28,
since
28 = 1 + 2 + 4 + 7 + 14.
The next two perfect numbers are 496 and
8128. These first four perfect numbers have been known to early Greek
mathematicians.
Several even perfect numbers are of the
form:
2(n-1)(2n-1)
where n represent some selected positive
integers. It should be noted that not all values for n gives perfect
numbers. Only those values of n where (2n-1) is a prime number,
would the above relationship produce a perfect number. A prime number of
the form (2n-1) is called a Marsenne's
Prime.
Perfect numbers are also triangular
numbers. This means that it is equal to
1 + 2 + 3 ..... + k
where k is a natural number. For
example,
6 = 1 + 2 + 3
28 = 1 + 2 + 3 + 4 + 5 + 6 + 7
496 = 1 + 2 + 3 + ... + 31
8128 = 1 + 2 + 3 + ... + 127
Even perfect numbers (except 6) also have
the property that they are equal to the sum of the cubes of consecutive odd
numbers starting from one. For example,
28 = 13 + 33
496 = 13 + 33 + 53
+ 73
8128 = 13 + 33 + 53
+ 73 + 93 + 113 + 133 + 153
Even perfect numbers, except six, also
have the property of
9n + 1
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