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 Previous Topic Next Topic 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 Previous Topic Next Topic

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