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PERFECT NUMBERS

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

 

More Mathematical Recreations

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