8128 AND PERFECT NUMBERS
The number 8128 is the most notable perfect number. 8128 = 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 A perfect number is a positive integer that is equal to the sum of its proper divisors not including the number itself, e.g. 28 is a perfect number because 28 = 1 + 2 + 4 + 7 + 14. The numbers 1, 2, 4, 7, 14 are the divisors of 28 i.e. they can divide 28. First perfect number is 6, i.e. 6 = 1 + 2 + 3. Other examples include 496, 8128 etc. It is not discovered that whether there is any odd perfect number or not. 8128 is related to the mersenne prime 127 127 = 2 7 – 1 2 7 – 1 is prime, then 2 7 -1 (2 7 – 1 ) must be a perfect number, i.e. 2 7 -1 (2 7 – 1 ) = 8128 Prime numbers of the form 2 p – 1 are called mersenne primes. Example include 3, 7, 31, 127, 8191, 131071, 524287 etc. Largest known prime number 2 74207281 – 1 is a Mersenne prime. Other features of 8128 are : 8128 is 127 th triangular number 8128