**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)}(2^{n}-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 (2**^{n}-1) is a prime number,
would the above relationship produce a perfect number. A prime number of
the form (2^{n}-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 = 1**^{3} + 3^{3}
**496 = 1**^{3} + 3^{3} + 5^{3}
+ 7^{3}
**8128 = 1**^{3} + 3^{3} + 5^{3}
+ 7^{3} + 9^{3} + 11^{3} + 13^{3} + 15^{3}
**Even perfect numbers, except six, also
have the property of**
**
9n + 1**
**More Mathematical Recreations** |